FLYBACK DIODES:
So it seems that the diode bridge will probably work IF the diodes are very fast and low forward voltage (like schotkey diodes). The current path for the flyback is also very strange in some cases. I am unsure if you can find a schotkey diode bridge, but you shouldn't use regular diodes. You need at least high speed diodes. But you should know that your circuit relies on a transistor being on to give the flyback current a path to flow. This means that if all transistors are off (like when you just decide to shut everything off) and flyback exists, then something will be destroyed.
I still suggest you switch to this for the diodes just because it's more common and easier to analyze.
https://www.modularcircuits.com/pic/simple_bridge.gif
It also allows a path for the flyback current to flow if all transistors are off for whatever reason.
You also seem to be using the diode bridge though for the voltmeters. What are those for?
GATE DRIVER:
D1 and D2 are not needed becuase the NPN BJTs already have such an intrinsic diode in them (double check me on this). But this is if your circuit does work (and I don't think it does).
Let's just look at the left side driver circuit because they are the same and everything that applies to the left driver also applies to the right driver:
-If Q7 turns on, both Q5 and Q6 stay off because Veb ~= 0 for Q6 and Vbe < 0 for Q5.
-If Q7 is off, then Q6 stays off because Veb < 0 for Q6 and I am uncertain what Q5 will do- it may or may not turn on.
The problem? Q5 and Q6 are of the opposite BJT types so the base-emitter voltages don't match up properly to turn them on with the way you have them connected. If you change Q5 to be PNP (since it is on the high-side, cloest to +V) and change Q6 to NPN (since it is on the low-side) it should then become a push-pull configuration that can be turned on. For example, you seem to realize that PMOS transistors are easier to switch when they are on the high-side, and NMOS transistors are easier to switch when they are on the low-side since that's what you did with Q1, Q2, Q3, and Q4. It's the same thing with NPN and PNP transistors. PNP is easier to switch on the high-side and NPN is easier to switch on the low-side, but for some reason you flipped it all around in your gate drive circuit.
I don't know what D1 and D2 are there for but I do not think they are needed (unless you can explain to me why you have them there).
THat is all I have time for now. I have to get back to studying, but you probably also want large capacitors across the power rails. I don't know if you included those in your circuit or not, and you may need some base resistors for your driver circuit in the same way you have them for the MOS gates. Gate resistors will slow down the turn on time for MOSFETs, but you might cause short-circuits if you don't have base resistors with BJTs.