It was definitely sung by Elves. The tricky part about these more specific questions is that we don't really know that much about the song.

Was it first sung in Aman?

Was it first sung in Aman - I bet so! :p

Is it Ode on some well known Lady?

Is it Ode on some well known Lady? - I wouldn't say so.

Is it the Aldudénië, or Lament for the Two Trees, by Elemmíre of the Vanyar?

Aldudénië - no, but very warm!

Narsilion, the song of Sun and Moon?

You got it, Pitch!!!

Wow, that was pretty esoteric. Nice though.

OK, ask away.

Thanks! I didn't think it would be that difficult! I saw somewhere in this thread things like languages, so I thought a song will be acceptable.

Person?

Nothing wrong with a song, that's just a little unusual. But a song that's only mentioned once in a single sentence of the legandarium, that's esoteric! If Tuor hadn't had the brilliant idea of asking whether it was sung in Aman, I'd still be guessing.

Person? - No.

Place?

Inanimate object?

Person? - No.
Place? - No.
Inanimate object? - Yes.

Was it a heirloom?

Was it elven made?

Person? - No.
Place? - No.
Inanimate object? - Yes.
Was it a heirloom? - Don't think so.
Was it elven made? - Yep.

Was it made by one of the Noldor?

Person? - No.
Place? - No.
Inanimate object? - Yes.
Was it a heirloom? - Don't think so.
Was it elven made? - Yep.
Was it made by one of the Noldor? - We're not told.

Was it made in Aman?

Was it a weapon?

Person? - No.
Place? - No.
Inanimate object? - Yes.
Was it a heirloom? - Don't think so.
Was it elven made? - Yep.
Was it made by one of the Noldor? - We're not told.
Was it made in Aman? - No.
Was it a weapon? - No.

Was it jewelery?

Was it jewelery? - No.

Is it a thing that you build? (eg tower?)

Was it given as a gift?

Is it a thing that you build? (eg tower?) - No.
Was it given as a gift? - Yes.

Phial of Galadriel?

Phial of Galadriel? - No (but you're close).

Was it given as a gift to someone in the Fellowship?

Was it the elven rope?

If it's not that, is it the belt she gave to Boromir?

Was it given as a gift to someone in the Fellowship? - Yes.
Was it the elven rope? - No.
If it's not that, is it the belt she gave to Boromir? - No.

The box she gave to Sam?

The box she gave to Sam? - Yes, that's it. Take it away, Evisse!

Cool!

Ask away :)

Is it a person?

Is it a person? - No

An object?