The datasheet of the IR2110 shows this as the typical connection:
And in some cases even a gate-source resistor is used, as in this and this post
But the app note shows this as the connection for a buck converter, no resistance whatsoever:
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.
Gate resistors should be used for input protection, the mosfet gate is a really thin layer of insulation sometimes nm's (tens to hundreds of atoms) thick. This creates a capacitor with some resistance. The problem is this layer can be blown away if a low enough impedance source is used, the ramp up time can damage the layer. ESD can also easily blow right through this layer (it doesn't take much joule heating in the material to create a hole because there isn't much material to start with and kV's of ESD create a lot of heat) The is really dependent on the mosfet, and on how much you care about protecting the input. There are many mosfets that have some kind of input protection. If they don't have any, you'll need it.
The IR2110 datasheet's recommendation in the example schematic has limiting resistors, I would put them in. With other drivers, I would check for some kind of limiting if they don't recommend them. DC to DC converters need to be very fast, and I don't know what the circuitry is but I'm willing to bet they limit the inrush current to cgs by design.
The gate resistance influences the Mosfet turn on and turn off time. So the value or its absence depends on the Mosfet gate capacitance and the desired on/off time.
No, don't use it! It's lamer's way to reduce EMI. The real way is to ensure short trace, close return path, etc. With resistor your rise and fall time will be longer and it will heat the MOSFET. Why would you do it? Even 10R may seriously harm your efficiency. Current limiting must be integrated in the gate driver.