As to your first question, documentation that can be use to prove your residence are bills for utilities, telephone, cable, and all other correspondance mailed to you at your new condo. I believe your tax preparer will ask that you provide a copy of your final settlement statement. I'll recommend you talk to your tax preparer ASAP to check on deadlines. Congratulations on your new condo.
Go to this site, http://www.ftb.ca.gov/individuals/new_home_credit.shtml, and check out the rules and availability for the California Home Buyer Tax Credit.
I hope this helps you.
If for some reason you need to prove that you live in the condo then I would think that if you have the utilities in your name and perhaps a few more bills sent to that address with your name on it that should help. You might also be asked to provide things like bank statements and payroll checks showing that you have done a change of address hence that you live there.