"I'd like to review the paperwork before signing". Usually when you start a new job you have to sign paperwork with the salary and other comp. It's totally reasonable to ask to review that before signing. It's also totally reasonable to go back to them afterwards and tell them you weren't happy with the offer after reviewing it.
I've never had this problem; usually it's either written in an email (rarely) or an official offer letter attached as a PDF (90%+ case). It's not at all unusual to ask for an offer in writing, and I'd never consider one real at all until I had that!
Yes but I've only received it in writing usually only after verbally accepting the offer, not before to play off of each other.