1) We travel on average once a month, with some months being heavier for travel than others of course.
2) During layovers, we generally try to find a meal or otherwise a relaxing place to wait until our connecting flight.
3) Even when traveling overseas, we make an effort to use the hotel gym to do a combination of cardio training and weight lifting.
4) While I'm not sure I'd use an airport gym, I'd want the same things available that I rely on at gyms otherwise (cardio and weight lifting equipment), with the added caveat that I would not want that gym space to be crowded.
5) I'm usually tired when flying, so that would greatly impact my willingness to use an airport gym. Also our layovers usually aren't super long (90 minutes to 2 hours), so the time it would take to work out (and shower . . . a gym would have to have shower access if I was to use it at the airport as I can't just drive home and shower) would make it unlikely that I'd use such a gym. This is all the more so as we're also generally looking to take advantage of a meal during layovers.
6) See above.

Honestly, I think this may be an example of an idea that sounds better on paper than it would be in practice. Wishing OP all the best in this endeavor, though.