I only heard three songs during the film, can anyone tell me where to get these or who does them?

There will not be a release, it was confirmed on twatter

Taya's Theme
Written by Clint Eastwood
Performed by Gennady Loktionov

Someone Like You
Written and performed by Van Morrison
Courtesy of Exile Productions, Ltd.

The Funeral
Written and performed by Ennio Morricone
Courtesy of Universal Music Ricordi srl.

American Sniper (2014) - Soundtracks - IMDb (http://www.imdb.com/title/tt2179136/soundtrack)

Taya's Theme: https://www.youtube.com/watch?v=otfj9VcZjSw