Yes, it's correct. The apostrophe tells me there's a missing letter - i. It's much more natural to say it this way and you should write it this way if you're being informal.

I think here it serves as a possessive pronoun: My sister's partner is a teacher. ( The partner of my sister ) It would be better to use pronouns with short form: My sister is older than me. She's a teacher.

Yes. When speaking we usually say "sister's". In writing you can do it either way, though for clarity it works best to say "sister is".

Yes, we generally use the short form unless it's strictly formal.