Yes, you can download and install TurboTax Desktop on up to five computers. Refer to the TurboTax Help articles Can I install TurboTax on more than one computer? and How many tax returns can I file with TurboTax?
If you purchase Deluxe directly from TurboTax it's the combo version that includes Federal plus a state program download to prepare state returns for 1 state. If you are purchasing elsewhere, be aware that Deluxe comes in 2 versions--one that is Federal-only, and one that a combo Federal and State, so you have to be careful in that regard.
If you do have any state returns and choose to efile, there is a $25 state efile fee for each (except no fee for NY state). There is no additional fee if you choose to print, sign, date, and mail the state return.