You can open a bank account with any bank anywhere in Canada. You don't even have to wait for your SIN card to do so. When I opened a bank account, I didn't have my SIN card yet.
They just want to verify your identity. So your expired passport should be fine for that purpose. Just in case, bring also PR card and Canadian driving license if you have one.
It takes a few months for CIC to process the test result and send you an oath ceremony letter. So don't worry, you just have to wait. CIC is very good at making people wait!
If your father registered your birth in Canada, then technically you are good to go. There might be some red tape/ fine prints involved, but you should definitely apply!!
You can join any political party in Canada as a permanent resident. As a PR, you can't vote but since you are going to become a citizen soon :) you will be able to be an active NDP member and vote for them. Good luck on your exam!