Can I use my last paycheck stub from last year to do my taxes until I get my W2?

You can use the pay stub for informational purposes on the tax return.  But you Must change that information when you receive the W-2 from your employer and before the tax return is filed.

You have to wait for the actual W2.  It usually doesn't match the paystubs.  Several deductions can affect the gross amount etc.  And don't be in a rush to file.

Turbo Tax will let you submit your  efile now but they will just sit on the Turbo Tax server until the IRS opens Efiling in late January.  Then it will take a couple days for the IRS to Accept or Reject it.    So don't be in a rush to file.  You can't get it back  if you need to change something or to add another form you get in the mail you weren't expecting.

That depends what you mean by "do" your taxes.  Prepare a test return? Yes.  As long as you update the numbers when you W-2 comes.  Actually file a return?  No.