Should I be submitting all forms that are printed out from TT state tax return? Do I also need to include the state W2 form as well as the cover sheet doesn't state that?

Yes, if you are printing and mailing in your tax return, you need to submit all the state forms that TurboTax says you need to mail. 

Generally, you do need to mail in your state copy of your W-2 form, but some states don't require that. The cover sheet should be accurate as to what you need to attach.