1. Yes
2. Yes at the CBP check they will ask for your passport and i797A. They know about AVR. If they are not sure they will go inside and check with thr manager before saying no to you. So avoid giving directions to them that you want to use AVR option unless you really have to.
3. Yes you will need to give a local address where you would want them to send your PR card. It takes goods 3 to 4 months, so any address you give ensure that your friend or family is going to be thr for good 4-5 months. I do not know about the UPS mail box rental. One more things- Address can be updated by calling or mailing them.
For me when doing PR activation - The only two docs they asked is the COPR and Passport. They asked if i am bringing anything to Canada or want to bring anything in future. I said no to both. If you want to then i guess you will need to have the list of goods doc for both. I tried to avoid any complexity as i was looking to just complete soft landing.
If you read the addendum they have sent along with COPR, they advise to keep all your documents with you just in case - previous all i797s, US DL, US SSN, India PAN, India Voter ID, India Aadhar, birth certificate, marriage certificate, Educational documents. Most probably no one will ask for it, but just in case. Previous i797s I guess may be asked for when you are returning to US at the CBP check post as well, so good to keep that handy.
At CPB checkpost while returning to US- They asked for only two docs- Passport and current approved i797A. They didn't even asked anything. My i94 was updated with i797 expiry date as well.