USCIS and SSN are separate so it wont show up there. Not sure if there is anywhere you can check but you can go to a social security administration office and enquire.
Regarding credit card, not sure if its dif for each bank. But I got mine with Chase since I already had a Chase bank account, just had to ask for it and that was it. Was just an online form with 2-3 questions about salary etc.
You also dont have to wait to start working. From the SSA website.
Do I need to have my number before I start working? "We don’t require you to have an SSN before you start work. However, the Internal Revenue Service requires employers to use your SSN to report your wages. While you wait for your SSN, your employer can use a letter from us stating you applied for a number. Your immigration documents can prove your authorization to work in the United States."
Heard plenty of people have been easily able to open an account with Bank of America with minimal info. So no need to wait around for your SSN before you start doing everything you listed