Which TurboTax payment method do I use having paid federal taxes through payUSAtax?

If you have already paid your taxes using one of the IRS's official payment processors outside of TurboTax (or you plan to make the payment using their site), then select "pay by check" and you can just ignore the payment voucher that prints up.  Selecting "pay by check" doesn't obligate you to pay by check, it simply produces the payment voucher for you to use.  You are free to then make the payment another way. 

In that case, do we still have to mail the return forms ourselves or does TurboTax do that?

We'd love to help you complete your tax return, but need more information. Can you please clarify your question?  Which return forms are you referring to? The payment vouchers or other forms from your return? Did you select to e-file your return or mail it in?

Asking about the tax return forms (1040 and such). I understand that I will have to mail in the payment voucher with my cheque.

Not sure if I selected e-file or mail in. Is there any easy way to check now?

This is what I see when I login to my account.