Need help calculating resistors for a modular model particle accelerator
Let's start by figuring out what resistors are needed, before settling on rough values (exact values aren't needed).

Resistors:


-R2 Is to limit the current into the transistor and make sure Vbe will be greater than Vce

R2 is there to give the phototransistor something to 'work against', and to source current into the following stage when the phototransistor is off. It doesn't need to source much current, as it's followed with amplifier stages. So no need for anything too small.

-R3 Is to limit the current through the base of the NPN transistor

This would be the case if driven from a low resistance output stage. However, the source current of the stage driving it is already limited by R2. You could omit R3. If you want to remove the phototransistor and sometimes drive it directly for debugging, then by all means have a nominal value there, 1k, 50k, whatever.

-R4 Is only there as most examples online had an emitter resistor, it may not be required

R4 is not required, as we're doing logic here. It's often used for linear amplification.

-R5 Is to limit the current through the base of the PNP, and into the collector of the NPN

Required. We'll get a value later.

-R6 Is to stop the PNP's base from floating, it may need to be lower to ensure that the NPN has enough collector current

Not wanted. You'd see this in a linear amplifier, but we're doing logic.

-R7 Is to limit the current into the PNP emitter, however it might be slowing down the switching speed of the Mosfet

Not wanted. Used in linear amplifiers.

-R8 Is to discharge the Mosfet gate capacitance, thus decreasing turn off time

It doesn't 'decrease turnoff time', it's the only thing that's doing any turnoff, as the driver transistor only sources, not sinks current, and the FET gate doesn't sink current either. As it's a fairly slow application, you'll get away with a resistive pulldown here, but any serious frequently switching application would need an active transistor to remove charge from the gate.
Use as small a resistor as you can, subject to the drive transistor being OK for the current. BC327 has rated minimum Hfe up to 300mA, so you've plenty in hand. A 300 Ω resistor would dissipate about half a watt from 12 V. Although you intend only short pulses of output, make sure that this resistor is rated for continuous dissipation, it's one fewer things to fail if your pulse stays 'on' accidentally for a long time.
Now we can go backwards and get some values.
R5 needs to turn the transistor on into 300 Ω. Let's use a very conservative Hfe for the second transistor of 30, so R5 becomes 10 kΩ.
Let's not make R2 and R3 (if used at all) too big by doing Hfe sums, just make them 10 kΩ as well.
Some folks put base-emitter resistors on their switching transistors to divert any leakage base current to get a good 'off'. These are only really warranted at high temperature or humid environments, you're unlikely to need them.
Sometimes you see a small resistor in series with the gate of a FET, often 10-100 Ω range. This is only needed in serious applications, where the FET is being driven by a low impedance driver, and it's to limit the gate charging current to limit ringing and overshoot. Will not be a problem in your application.
Is the driver adequate? This section is for the OP, when he's got a lot more under his belt, and people who might want to downvote what may appear an excessively hand-wavy answer, so they don't need to look up the FET and do sums. The FET has a 100uS SOA going right out to 40 V and 300 A, so as long as all switching is done within that time, there's no need to worry. The FET needs about 100nC gate charge to get through the Miller plateau, and then some. Assuming Hfe=100, Rs=10k will source 1 mA to base, so 100 mA at the collector. 100 nC at 100mA = 1 us, so switch on is OK by two orders of magnitude. It's a logic level FET, so Vgs(Miller_plateau) is low at 3 V. Even so, 300 Ω R8 will abstract charge at 10 mA, or in 10 us, still an order of magnitude for safety. The FET only has an unclamped inductive energy rating of 320 mJ, whereas the inductor can store 1 Joule at full current. That clamping zener is essential therefore. Note the FET only goes to 40 V, so the zener must come in before that. Maybe a higher voltage FET would be a good investment.