1. I should be using my parents' current address since my passport address won't work. Correct?
I tried doing this and the did not accept parent address proof at "Vijay X road Passport Seva Kendra". I even met with the RPO at Ahmedabad office and they said, you have to have Indian address in order to complete this
2. Will Aadhar having a different address be a problem? Should I try getting it changed before my visit?
Yes, that's what I did. It will take 3-4 days to update address for aadhar card and then you can print it to any Internet cafe and use it as a proof
3. When should I start the application given that it automatically assigns you an interview date at the PSK?
I'd say 2-3 days in advance, as you will be visiting Ahmedabad office (?) and that is my experience with them that they have appointments available in next 2-3 days.

Hope that answers your questions!