1) No, contract doesn’t mean it has to be on paper. It’s a verbal contract
2) As for the words, you can read what needs to be said here:
3) The dowry doesn’t need to be paid at the time this is pronounced. It can be paid later.
If she’s virgin and she’s not fully independent (meaning she doesn’t make her life decisions independently), then Ayatollah Sistani says her father’s permission is required. Yes, if he’s not involved in making any decisions for her (he’s left it up to her to marry who she wants when she wants, for example), then his permission is not required
Now this is all assuming her father is Muslim. But if her father is not Muslim, then his permission is not required. She may do the marriage contract without his permission or knowledge.