Should I use a resistor between the gate driver and MOSFET (gate pin)?
Maybe. The MC34152 datasheet pp.8 on shows a series Rg to damp oscillations, and reverse-bias Schottky diodes for catching negative ringing spikes at the driver. Wouldn't hurt to have these in your layout. You could stuff zero-ohm for Rg if you don't need damping, and no-stuff the diodes if you find the ringing isn't too bad. Have one resistor/diode per FET, don't share them. Place them near the gate.

No pull-down is needed at the gate drive. But you will want a pull-down on the driver input to make the default state 'off'.

Also, while we're discussing the inputs - tie them together and use both of the separate outputs, one for each FET. The way you have it - driving 2 FETS together - kind of defeats the purpose of the buffer.

Finally, if your motor is a normal brush type you will want to use a freewheel diode across it to catch the flyback spike when the switches turn off. For BLDC this isn't an issue. Yes, C2 does this too, but the diode is better.