Chances are no. The banks have a computer system ( names vary by type of bank) where they run your information in before setting up an account. You owing another bank money will show up and they may not open an account for you at the new bank.
You can try, but don't be surprised if it does not work.
Generally you can. Having credit products in other banks would only matter in case you would like to apply for another bank's credit products (all banks are interested your debtedness for risk assessment reasons)
In my personal opinion, I owed one bank money on overdraft fees and decided to just open up another account elsewhere. When I went to the bank they asked me for my personal information and I guess if you do owe other banks money or something that they are able to pull that information up. In the end I had to pay for those overdraft fees to open a new checking account with a different bank.