On the IGBT data sheet, the second section labeled IGBT shows the VGE (th) threshold voltage max is 6.2V, so, yes, 12V will turn them on sufficiently.
However, just below that, Rce (resistance of the collector to emitter) and VCE (sat) are both spec'd at Vge of 15V, so the on resistance and the saturation voltages may be a little higher, which means the gates will dissipate more power and need a larger heatsink to drive max current through them.
The data sheet shows a VGE leakage current of .1 to .3mA (VGE = 0V), and the 4047 will output approx. 2.4mA, and there is no other Ige listed for on value, so it's probably safe to assume this part will drive the gates ok. The resistors are good to stop ringing noise in the gates as they have a high capacitance value, although 470 might be larger than needed. The best value will be found through experimentation in the lab.
I don't know what frequency you're trying to obtain, probably 50Hz? But I don't like RC's for frequency generation. They have a large drift over temperature. Just my opinion...
The inductor? That would depend on the switching frequency and the inductance of the transformer...