1. I can read books for the elderly people in nursing homes.

2. I can read books to the elderly people in nursing homes.

(Are both OK?)

3. I can sing old songs for the elderly people in nursing homes.

4. I can sing old songs to the elderly people in nursing homes.

(Are both OK? Is there any difference between them because of the preposition?)

All are OK, yes. #s 2 and 4 are probably most accurate, but all four sentences work just fine.