There are many of those.

TYRN GORTHAD: Thank you, nurse!’ (American text speak). Here, a twisting cave leads to the beginning of the underworld.
H: Nimble Field Marshal gains direction even as all becomes confused here.
ERESTOR: See him before the decay returns; a sibilant is swallowed.

FORN: Ask Debussy when it sounds like he sings. It might be after noon.
ERIADOR: Thingol’s land is transformed into another, as a fricative element gives way to hesitation in the general turmoil.
LINDIR: Is he a lover of confectionery, with just one small exception?
LEMBAS: Food guilt? It’s confusing when someone lays that on you.
ODO PROUDFOOT: His table manners leave much to be desired.
WITHYWINDLE: Shrivel without hesitation, here in France? Decline, after first loss. But on it flows.
SILVERLODE: Fifty pence pieces can really weigh your pockets down, I hear! That considered, on it runs.
HOBBITON: Here you’ll find a stove with tooth marks in it, they say!
IMLADRIS: Girl traps boy here. Mother too, initially, and first.
PARTH GALEN: Here find oriental bread, lose two articles, and meet a physician.

OLD WINYARDS: Flatulent and sibilant children’s author? (Forget his surname). It's confusing, but perhaps he’s sampled this?
FANUIDHOL: Spanish greeting for Balin’s directionless dad; despite the confusion, it’s not that far from where his son lies.

TINDROCK: Charlatan of a professor – looks peaky after much confusion!
HALFAST: He’s semi-sedentary, with very minimal disturbance.
EVERARD (TOOK): An eternally tough dancer, we hear.

RAUROS: Dark Lord changes direction as he tumbles here.
I DRANN: What Drogo might have yelled here, in 2980, had he been a Cockney?
NOB: Does he really ban the use of a certain plosive?
GILDOR: To a linguist’s mind, he seems almost to specialise in golden tautology.

Well, if I google Field Marshal he comes up straight away. Maybe that's because I'm googling in the UK, but he was very famous. A little WWI research would quickly unearth him.

Haig?

Haig it is.

(Not Hair, as Autocorrect thinks).

It was right there in front of me all along and I couldn't see it....


Hithaeglir (HAIG+LITHE+R)

TYRN GORTHAD: Thank you, nurse!’ (American text speak). Here, a twisting cave leads to the beginning of the underworld.
HITHAEGLIR: Nimble Field Marshal gains direction even as all becomes confused here.
ERESTOR: See him before the decay returns; a sibilant is swallowed.

FORN: Ask Debussy when it sounds like he sings. It might be after noon.
ERIADOR: Thingol’s land is transformed into another, as a fricative element gives way to hesitation in the general turmoil.
LINDIR: Is he a lover of confectionery, with just one small exception?
LEMBAS: Food guilt? It’s confusing when someone lays that on you.
ODO PROUDFOOT: His table manners leave much to be desired.
WITHYWINDLE: Shrivel without hesitation, here in France? Decline, after first loss. But on it flows.
SILVERLODE: Fifty pence pieces can really weigh your pockets down, I hear! That considered, on it runs.
HOBBITON: Here you’ll find a stove with tooth marks in it, they say!
IMLADRIS: Girl traps boy here. Mother too, initially, and first.
PARTH GALEN: Here find oriental bread, lose two articles, and meet a physician.

OLD WINYARDS: Flatulent and sibilant children’s author? (Forget his surname). It's confusing, but perhaps he’s sampled this?
FANUIDHOL: Spanish greeting for Balin’s directionless dad; despite the confusion, it’s not that far from where his son lies.

TINDROCK: Charlatan of a professor – looks peaky after much confusion!
HALFAST: He’s semi-sedentary, with very minimal disturbance.
EVERARD (TOOK): An eternally tough dancer, we hear.

RAUROS: Dark Lord changes direction as he tumbles here.
I DRANN: What Drogo might have yelled here, in 2980, had he been a Cockney?
NOB: Does he really ban the use of a certain plosive?
GILDOR: To a linguist’s mind, he seems almost to specialise in golden tautology.

I thought I had used 'agile' as the synonym for 'nimble,' but you're right. It took me by surprise when you posted it, but yes, I must have used 'lithe.'

Well done. Over to you!

Inspired by Pervinca's puzzle:


1. Initially, electronic area is here.
2. Bring over, in two directions. Confused? It's here.
3. Poisonous plant grows in clump here.
4. A mathematical term? Here?
5. Here is a country of endless sand hills.
6. He is a confused writer.
7. Hurry, an endless approval returns here.

1. Ea

4. DUNLAND?

Dun(e) + land.

Maybe ERIADOR for the password?

7. RUSHEY, a village in the Eastfarthing?

Rush + ye(s) backwards.

...

As all but 6 seem to be places, maybe the theme is GEOGRAPHY and 6 is a cartographer - a writer of maps? ;)

2. RIVENDELL.

Deliver + N + L.

Nope. Not cartographer, but Galadriel55 is close with Ea. The character in 6 is theorized to be Ea.

I assumed you meant 5 for Dunland, not 4.

E: Initially, electronic area is here.
RIVENDELL: Bring over, in two directions. Confused? It's here.
I: Poisonous plant grows in clump here.
A: A mathematical term? Here?
DUNLAND: Here is a country of endless sand hills.
O: He is a confused writer.
RUSHEY: Hurry, an endless approval returns here.


For 1, 'initially' applies only to 'electronic' part....

1. EREGION

E + REGION for area.

I *loved* G55's answer, though!

EREGION: Initially, electronic area is here.
RIVENDELL: Bring over, in two directions. Confused? It's here.
I: Poisonous plant grows in clump here.
A: A mathematical term? Here?
DUNLAND: Here is a country of endless sand hills.
O: He is a confused writer.
RUSHEY: Hurry, an endless approval returns here.

I'll try ANNUMINAS for A, but it only really accounts for A + MINUS. 'A new minus' seems to stretch the function of the question mark a bit too far.

And I want to try IVRIN for I, because it contains two-thirds of IVY, which I believe is poisonous. But everything else so far is Third Age, which gives me a second reason to believe I'm wrong.

Ah! Is O ORALD, another name for Tom Bombadil, who I think might be the spirit of Arda, or something? (It's also an anagram of ROALD).

Maybe IVY BUSH (the tavern) for I? A bush isn't really a clump, though ... and I guess there are hundreds of poisonous plants besides ivy.

Or is it IMLAD MORGUL, and does it simply have poisonous plants? Is this the place where there are flowers, beautiful and yet horrible of shape?

EREGION: Initially, electronic area is here.
RIVENDELL: Bring over, in two directions. Confused? It's here.
IVY BUSH: Poisonous plant grows in clump here.
A: A mathematical term? Here?
DUNLAND: Here is a country of endless sand hills.
ORALD: He is a confused writer.
RUSHEY: Hurry, an endless approval returns here.

Annuminas isn't in Eriador. ;)

I thought it was ... wasn't it a place of the Dunedain in Middle-earth? And possibly also a place in Numenor?

Not the place I was thinking of.

I am looking for a place which shares the whole name with a mathematical term, not just a part of a name.

It's listed here: http://tolkiengateway.net/wiki/Category:Eriador

under A.

Archet?

(Maybe it's a kind of arc?)

I'll check out your link ....

... ANGLE.

EREGION: Initially, electronic area is here.
RIVENDELL: Bring over, in two directions. Confused? It's here.
IVY BUSH: Poisonous plant grows in clump here.
ANGLE: A mathematical term? Here?
DUNLAND: Here is a country of endless sand hills.
ORALD: He is a confused writer.
RUSHEY: Hurry, an endless approval returns here

Nicely done. Over to you.

I enjoyed that, Urwen. :)

Here's another one:

1. Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
2. Half German article and liquid marketing. For him.
3. Place of mirth ... but it's not unmixed.
4. Brag - dubiously, at first - but chaos ensues to expose him.
5. Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
6. Not all of these are evil.
7. Does one hesitate to suggest a pub here?
8. She's inflamed - and troubled.
9. One must dress, groom, prepare? - put nails in order, certainly - for them.
10. Is David's foe around here?
11. Beat a Weasley? He's there - but discombobulated.
12. Group of sheaves disturbed by - or rather for - the first ones to kill.
13. They're beardless and bewildered.
14. A role reversed; an article swallowed; a girl revealed.
15. Merry and Pippin, in white satin? (For thus it sounds, in song).
16. A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
17. See this fearsome thing return, endlessly crazy, to endlessly deceive.
18. Aged apparatus collapses to reveal him.

8. Rose (SORE)
10. Osgiliath (IS + GOLIATH)

14. Elanor (ROLE backwards + AN)

A hat trick!
 

1. Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
2. Half German article and liquid marketing. For him.
3. Place of mirth ... but it's not unmixed.
4. Brag - dubiously, at first - but chaos ensues to expose him.
5. Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
6. Not all of these are evil.
7. Does one hesitate to suggest a pub here?
ROSE: She's inflamed - and troubled.
9. One must dress, groom, prepare? - put nails in order, certainly - for them.
OSGILIATH: Is David's foe around here?
11. Beat a Weasley? He's there - but discombobulated.
12. Group of sheaves disturbed by - or rather for - the first ones to kill.
13. They're beardless and bewildered.
ELANOR: A role reversed; an article swallowed; a girl revealed.
15. Merry and Pippin, in white satin? (For thus it sounds, in song).
16. A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
17. See this fearsome thing return, endlessly crazy, to endlessly deceive.
18. Aged apparatus collapses to reveal him.

The Return of the King for the password?

Verily
 

T: Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
H: Half German article and liquid marketing. For him.
E: Place of mirth ... but it's not unmixed.

R: Brag - dubiously, at first - but chaos ensues to expose him.
E: Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
T: Not all of these are evil.
U: Does one hesitate to suggest a pub here?
ROSE: She's inflamed - and troubled.
N: One must dress, groom, prepare? - put nails in order, certainly - for them.

OSGILIATH: Is David's foe around here?
F: Beat a Weasley? He's there - but discombobulated.

T: Group of sheaves disturbed by - or rather for - the first ones to kill.
H: They're beardless and bewildered.
ELANOR: A role reversed; an article swallowed; a girl revealed.

K: Merry and Pippin, in white satin? (For thus it sounds, in song).
I: A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
N: See this fearsome thing return, endlessly crazy, to endlessly deceive.
G: Aged apparatus collapses to reveal him.

Umbar for U.

K is... Knights?

T: Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
H: Half German article and liquid marketing. For him.
E: Place of mirth ... but it's not unmixed.

R: Brag - dubiously, at first, you hear - before chaos ensues to expose him.
E: Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
T: Not all of these are evil.
UMBAR: Does one hesitate to suggest a pub here?
ROSE: She's inflamed - and troubled.
N: One must dress, groom, prepare? - put nails in order, certainly - for them.

OSGILIATH: Is David's foe around here?
F: Beat a Weasley? He's there - but discombobulated.

T: Group of sheaves disturbed by - or rather for - the first ones to kill.
H: They're beardless and bewildered.
ELANOR: A role reversed; an article swallowed; a girl revealed.

KNIGHTS: Merry and Pippin, in white satin? (For thus it sounds, in song).
I: A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
N: See this fearsome thing return, endlessly crazy, to endlessly deceive.
G: Aged apparatus collapses to reveal him.

Havens for second H (from SHAVEN)

T: Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
H: Half German article and liquid marketing. For him.
E: Place of mirth ... but it's not unmixed.

R: Brag - dubiously, at first, you hear - before chaos ensues to expose him.
E: Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
T: Not all of these are evil.
UMBAR: Does one hesitate to suggest a pub here?
ROSE: She's inflamed - and troubled.
N: One must dress, groom, prepare? - put nails in order, certainly - for them.

OSGILIATH: Is David's foe around here?
F: Beat a Weasley? He's there - but discombobulated.

T: Group of sheaves disturbed by - or rather for - the first ones to kill.
HAVENS: They're beardless and bewildered.
ELANOR: A role reversed; an article swallowed; a girl revealed.

KNIGHTS: Merry and Pippin, in white satin? (For thus it sounds, in song).
I: A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
N: See this fearsome thing return, endlessly crazy, to endlessly deceive.
G: Aged apparatus collapses to reveal him.

Well done! I thought that was a slightly sneaky clue.

Second T: Not all TEARS are an evil.

Second E: A sailor has SEA LEGS, which in the singular becomes the majestic EAGLES.

hS

T: Here, a biblical fellow returns, to meet an archaic palindrome and a sibilant ocean.
H: Half German article and liquid marketing. For him.
E: Place of mirth ... but it's not unmixed.

R: Brag - dubiously, at first, you hear - before chaos ensues to expose him.
EAGLES: Seasoned sailors have more than one of these. But be that as it may, it's mangled to reveal these majestic creatures!
TEARS: Not all of these are evil.
UMBAR: Does one hesitate to suggest a pub here?
ROSE: She's inflamed - and troubled.
N: One must dress, groom, prepare? - put nails in order, certainly - for them.

OSGILIATH: Is David's foe around here?
F: Beat a Weasley? He's there - but discombobulated.

T: Group of sheaves slightly disturbed by - or rather for - the first ones to kill.
HAVENS: They're beardless and bewildered.
ELANOR: A role reversed; an article swallowed; a girl revealed.

KNIGHTS: Merry and Pippin, in white satin? (For thus it sounds, in song).
I: A painter's choice - Rembrandt's, initially - and Goya's, finally - mingles to reveal him.
N: See this fearsome thing return, endlessly crazy, to endlessly deceive.
G: Aged apparatus collapses to reveal him.

Sea legs indeed! I learned this term from watching The Onedin Line.

Some hints please?

F: Choose a Weasley and find a synonym for 'beat.'

I have a feeling that the 'Weasley' is Fred.

Not Fred.