To be clear, this hotel charges by the hour (or 4 hour chunks) and is designed for shorter 8-12 hour transits, so 22 hours could be very expensive. There aren't really facilities except for the room, breakfast, and a gym too - but the pool bans under 5 year olds, which may be a problem for you in terms of children's entertainment. You can email and get a quote, and current rules for usage/facilities/COVID closures.

In my opinion, with 22 hours you are best to go to a city hotel, and it will likely be cheaper to have one of those for two nights (or with a very early & late paid check-in/out) rather than spending 22 hours in the airport, which unless you are about to sleep for 20 hours or so. Note, I appreciate with a little kid this may seem less attractive, as you'd have to spend ~30 minutes in a taxi each way and perhaps (COVID restrictions dependent) do a PCR test to get into Qatar. However, the rooms and hotel are very much for a transit - it is basically a corridor and a comfortable room, my own kid would be very bored there - the Qatar lounges do have family rooms and play areas for children though, but not enough for 10+ hours of awake time.

If in your scenario I would likely do say: 1) Of course business class, as the lounge/experience on Qatar is excellent, 2a) Head into town and sleep (let's say you'd sleep about 5.30-6am), wake up for lunch, spend time by the pool at a hotel. 2b) Or if you wanted to "wake up" just before your flight, get to the hotel, have a nice breakfast, chill by the pool, wander around the souk/Islamic art museum, then sleep around lunchtime for 8 hours. In either scenario, arrange a late check-out to say 8pm, have dinner and then get to to the airport around 10-11pm for ~2 hours in the Qatar lounge before the flight.