Your employer SHOULDN'T know your debit card or credit card numbers. If you change the checking account where you have your pay direct deposited, you need to give them the new number so they can send your pay to the new account.
If you changed bank accounts, and have direct deposit, then your employer needs to know your new bank account number and the routing number. Just changing debit/credit cards but keeping the same bank account, you don't have to tell your employer anything.
yes otherwise the money won't be transferred to your new bank account.
If that is where you get direct deposit of your pay, yes.