I have TurboTax Deluxe. It includes "1 State product via download". Does that allow me to use that same state several times ie for more than one return?

Yes, the one free state download can be used for an unlimited number of returns (although you can only e-file up to 5 Federal returns).  If you needed another state, there would be an additional charge.

Note: There is a $19.99 charge per state return to e-file.  This can be avoided by printing and mailing the return.

So do u mean the same single free state is only for the free federal returns installed on the 5 computers?