Yes. I have 2 accounts with Chase myself
Your accounts are tied to your name and SS number so they will come up online or in the system. Open a separate account at another bank. It could be a high yield savings account online which I recommend because they have way better rates than regular banks. www.ratebrain.com has good rates. Make sure they are not promotional such as everbank, that is just for 3 months.
Yes, it is. Or at other banks.