I don't think that there is much difference internally between my H21 and your H22 except in the design of the plastic front to the case. The supplied rear exhaust fan seems quiet and cools the case effectively as far as I'm concerned. It is a 3 pin connector an I have it attached to a motherboard 4-pin connector on an Asrock B450 board with a Ryzen 2600 and a GForce 950. So similar but different to your build. I have 2 Arctic 4-pin 120mm fans in the 'roof' that are rather unnecessary, and I have left them unconnected for the time being, but they may get connected for a hot period in the summer (which lasts 3 or 4 days in the UK). The top of the case has countersunk holes in the right place for 120mm fans.
There are also fittings for two 120mm fans in the plastic housing at the front, which when it is removed reveals an inner mesh dust filter with clips, and the fans screw to that. It's a loose and rather unusual fitting.
I found that the case was a delight to install the system in, and the motherboard holes exactly lined up with the prefitted standoffs. The I/O shield clipped in as a better fit than any other case I've had. I had originally installed the system in a case I had for about 20 years, but there were too many drive bay 'holes' in it (I'd lost the fittings) and it was ugly. Also, the Versa H21 allows 3.5 and 2.5 inch devices to go in 'sideways' which is a good idea for access, and they are in the right place for a big graphics card which didn't exist when my original case was made. However, although the smaller clip-in trays are a snug fit, the larger 3.5inch device trays aren't well secured, and I had to secure them with cable ties (2 drives). The H21 is better looking than I expected.
One thing that irritates me with my H21 is that the connectors for USB3 and USB2 take up the motherboard connectors for 2 sockets each, but only give one socket on the front panel. You typically only get one USB3 socket on a B450 board, and so need an additional PCIe card with another internal USB3 socket if you install a front panel unit like I have done with a card reader and extra sockets. And when you do that, you may duplicate the front panel audio jack plug sockets! It may not bother you, but I have removable USB3 devices that I prefer to connect at the front.
As far as 'toolless' fittings for the 5.25inch drive bays, they are one side only - the other side needs conventional screws.