Locations of certain places

[Legate of Amon Lanc]

Well... that's a little bit complicated, but I'll try to answer as best as I can. Hopefully someone else can make the information more precise.

Angband is just outside the map in Silmarillion, I guess about the same longitude as the long vertical line is (maybe a little bit to the east. But it is big anyway). In the Silmarillion, we are told that:

Originally Posted by Silmarillion, Chapter 10 "Of the Sindar"

But Morgoth, as has before been told, returned to Angband, and built it anew, and above its doors he reared the reeking towers of Thangorodrim; and the gates of Morgoth were but one hundred and fifty leagues distant from the bridge of Menegroth: far and yet all too near.

(emphasise mine)
That's probably the closest description we have. You can measure it for yourself

Concerning Utumno, it is probably somewhere "around" (but the distance can be quite great), maybe a little bit more to the north and definitely further to the east.

Originally Posted by Silmarillion, Chapter 14 "Of Beleriand and Its Realms"

In the north of the world Melkor had in the ages past reared Ered Engrin, the Iron Mountains, as a fence to his citadel of Utumno; and they stood upon the borders of the regions of everlasting cold, in a great curve from east to west. Behind the walls of Ered Engrin in the west, where they bent back northwards, Melkor built another fortress [Angband]

In the Atlas of Middle-Earth by Karen Wynn Fonstad (quite a good tool, and I consider it quite trustworthy) Ered Engrin are pictured like some sort of circumpolar-oriented mountains (of course Middle-Earth was flat back then, so it's just half a circle). In that case Utumno lies practically on the pole (or, on the edge of the world) and Angband is in the southmost part of the mountains, somewhat more to the west than Utumno.

The last three are a problem, since they are all situated to the same environment (more or less, Cuiviénen is maybe a little bit better, but the two others are quite a big problem). At the time of the Lamps, when the first realm of Valar was on the island of Almaren, Arda looked completely different than after, because the wars between Valar&Morgoth even before the coming of the Elves completely changed its face. You simply had the world, where the only sea was in the midst, on the northern edge there was one lamp, soutwards the other, and in its middle there was the island where Valar dwelt. I guess if you sought for the place using geographic coordinates, you'll find the places matching some spot between M-E and Valinor (somewhere in the sea). Maybe the original island matched even the place of later Númenor?

And Cuiviénen. I'm not much sure about this one, but Fonstad in her atlas puts it about the same latitude as lake Rhun, but more to the east, as if you took the distance from Misty Mountains to Lake Rhun and multiplied it twice. She draws there the sea of Helcar, which, interestingly, is flooding Mordor and then all the space through Dagorlad to lake Rhun, and then further to the east as far as Cuiviénen. I'm not much sure about it being 100% reliable, but I think it will be more or less precise. We don't have much info about it, anyway, and Fonstad relied solely on the Professor's work when she was making it (but where he was not clear about something, she might have interpreted it in her view).