You are correct. The gate is high impediance once the switch is on or off, but during the transitions there is current flow to charge/discharge the capacitance of the gate. That's why you need a gate resistor between your driver chip and the discrete, to control the turn on/off speed. Look a the spec sheet and you'll see that a smaller Rg results in faster switching (but more EMI).
So as to controlling 3 IGBTs with one driver output. Yes it will work, but there may be issues with timing if that is critical in your circuit. If the IGBTs are different, or the gate resistors are, then they will turn on/off at different speeds possibly causing problems. Also if you are switching at very high frequency you need to make sure the driver chip can handle the power required.
If you're running below 10 kHz and all the IGBTs are the same, I would reccommend increasing the Rg from the spec sheet, but other than that it should work just fine.
