Can I open a bank account in Thailand with just a Non-B visa?

With a Non-B visa, opening a bank account should be easily doable. You'll no doubt need your work permit booklet (obtained by your employer from a government office) and of course passport, and an opening deposit. Since you'll be legally making money in Thailand, your employer will presumably need a place to deposit your salary. Find out from your employer which bank they prefer. They should send an employee with you to the desired bank branch and set up an account for you. I worked in Thailand for ten years until recently, and they did that for me.

Your employer should help you. They will need you to have a bank account so they can pay your salary. Someone from HR should be available to help you and they have a preferred bank and branch.

Get a residence certificate from Immigration and you should be ok

You can absolutely try