IRGB14C40L and MC33153
1) The continuous gate current according to the specs of the IGBT is 1 mA, the 10 mA is allowed in pulses of 1 ms with 100 Hz frequency. This makes me a bit hesitant to put 8 mA or so as with your suggested set-up. I'd propose a maximum of 5 mA, or R1 = 2.7 kOhm.
For faster discharge you could take R2 = 10 kOhm. For D1 you'd need a really fast switching diode with a low reverse recovery time.

2) In decoupling switching phenomena from the supply you could take a tantalic capacitor for C1. Make sure to obtain and properly read the data sheet in order to select the right series resistor, 1 uF should do it.

3) Yes, and also the driver has a 100..200 kOhm pull-down resistor that would act the same, although I'm not sure when you suspect the gate to 'float', except for when the circuit is switched off.

Regarding the resistor in series to a tantalum:
Tantalum capacitors have an extremely good high-frequency behavior and very low 'effective series resistance' (ESR) and are therefore excellent candidates to absorb disturbances due to switching. However, depending on the voltage and frequencies of the disturbances, the low ESR can lead to quite high compensation currents which can destroy the tantalum through overheating.
Once I also wasn't aware of this problem and repeatedly blew out the power stage of an experimental inverter that I've been working on. We repeatedly found a blown tantalum after the failure, started to actually read the specs and found the recommendation to apply series resistors.