Does the IR2110 need a gate resistor?
Short answer, yes you want a gate resistor.

Longer answer, it reduces EMI by slowing down the MOSFET which in turn reduces ringing and high frequency components of the switching waveforms. On negative side this causes more power losses as the on and off times are increased. You can see in one of the schematics that there's a diode across the resistor, typically switching the mosfet off can be done quicker without ill effects depending on topology.

None of these circuits actually seem to be SMPS circuits so unless there is fast switching involved, it may not matter. 

Gate source resistance is probably just to keep the mosfet OFF if circuit is not functioning so there's no 300V in output. In SMPS applications you may run into dV/dT induced spontaneous turn-on as mosfet parasitic capacitances conspire to raise gate voltage above threshold voltage. However in this instance a small cap is added from gate to source, not a resistor.