I wouldn't open a separate checking account, just talk to your bank and find out what protections you have in place. One of the differences with debit and credit cards is that credit cards almost always have the chargeback protection, while debit (checking) cards do not often have it. Sometimes debit cards do have the same protection though, it really varies from what I have heard. So check with your bank.
As for the fishing, yest that's a possible scam to get your email address, or it could even be someone accidentally entering your
email address to try and change their
password. Either way, you are correct in that the safe thing is to ignore the emails.