How long does it take once HR tells you that you are the finalist for a position and they would like to hire you (Big Corp)... until you actually get the contract with all the details and signed?
Once you go that stage, it normally goes very fast. From my experience they send you a contract by e-mail so you can confirm you accept its conditions, and the the hard copy by mail, together with a list of admin stuff you need to organize (I got the offer on Friday, confirmed on Monday, received the hard copy of the contract Wednesday.) For the other jobs I have, it was more or less the same.
If you have agreed to the offer verbally, I would suggest you to send an e-mail to the HR person whom you spoke with, thanking and stating the agreed salary and date of start.
And, by the way, congratulations for you new position and best of luck :-)
HR would inform him/her. Typically, once the decision is made to make an offer, the discussion on terms / conditions is between HR and the candidate. The hiring manager could get involved in case the negotiation stalls for whatever reason.
Can't hurt to let hiring manager know you signed and are looking forward to start of course.
Just let HR know and give them a slot to reach you on or have them propose a slot. Then just wait. Presumably you want to speak with them to clarify some things or do some negotiation on final points. They've given you a contract so are fairly committed. Chasing them will just dilute any negotiating power which you may have.