Of course you are totally free to make up needed words on the spot. This kind of ad-hoc language play is the innate right of all speakers.
The process follows certain rules, nevertheless. Let's take your example:
Friday nights are unwindy nights
At first read, without seeing your explanation, I thought this meant that Friday nights are characterized by a lack of wind. Here, "unwinding" would fit just right and obviate the need for a nonce coinage.
The usual thing is to flag or mark the novel use in some way, so I agree with your idea of "unwind-y," or potentially "unwind-ey" or something to alert the reader that unusual maneuvers are afoot. Unfortunately, "wind" (breeze) and "wind" (coiling) are homographs, so "unwindy" is unhappy.
Friday nights are when I unwind.
Friday nights are for unwinding.
Friday nights I unwind.
While it's always acceptable to push the envelope, language-wise, you have to consider the puzzle you're presenting the reader and offer sufficient clues to allow smooth decoding.