Can someone help me understand this H-Bridge?
Yes.
a) Yes. b) Possibly; see answer #4.
R1 limits the current for low-speed forward and reverse operation. T9 provides high-speed forward operation.
It raises the switching threshold for P1 by one diode drop. My guess would be that if you drive P1 and P2 high simultaneously (from a dead stop), this allows the low-speed circuit to get the motor started before the high-speed circuit kicks in, providing smoother acceleration.
The resistor-capacitor-diode combination provides a soft-start, fast-stop characteristic for the motor. When starting, the resistor and capacitor have a time constant of about 0.224 seconds, which helps prevent the motor from abruptly spinning the wheels. When stopping, the diode dischages the capacitor immediately for faster response.
See my comment on your question. When hooked up correctly, T1 feeds base current from the high-side PNP (Q2) directly to the corresponding low-side NPN (T5).
Only if you remove the three capacitors. You'll also need to add freewheeling diodes across each of the five power transistors.