When I was watching the GreatScott video on the series gate resistor, he made it seem neccessary.
Generally, gate resistor may serve two functions:
1. Slowing down gate charge / discharge
2. Damping of parasitic LC tank circuit
In many cases these two functions interleave and can be thought to be same, but I believe it is better to think of them as separate. Slowing down happens because the resistor limits the gate charge / discharge current. For example, in some cases you may use really high value resistors to achieve slow and smooth switching. You may also use very low value resistors to achieve maximum speed and low switching loss (reduce the time spent in linear region).
If you think about it in frequency domain, you may notice, that the speed at which gate is charged /discharged (rise and fall time) defines the frequency content of gate current waveform. For example, fast rise time = high frequencies involved.
The gate has capacitance plus the mosfet package, leads, PCB tracks has inductance. So it forms the LC tank, which no one wants, but it is still there (hence, generally called parasitic). This LC tank resonates if it is excited by currents containing high enough frequencies. So if gate is switched fast enough, parasitic LC tank will resonate and create ringing (and overshoot / undershoot). Here gate resistor not only lowers the frequencies involved, but also reduces the quality (Q factor) of this LC tank (damps it), so the energy dissipates and the resonance becomes weak. Generally, every combination of mosfet and PCB layout has an optimal value of gate resistor to damp the ringing enough (critical or near-critical damping) to achieve clean gate signal.
In your schematic, the gate driving source (peak detector circuit) is so very slow, that the ringing is impossible. The gate resistor is not needed, because it will change nothing and thus has no practical function. Hope, this makes sense.
Also, why not to the zener diode, and instead use the regular silicon diode? I'm a noob, so I'd love to know the reasoning.
First of all, I never saw a zener in your schematic, just a schottky diode. Anyway, Zero999 suggestion of a 1N4148 is a good one. It is a nice, fast, low capacitance widely available diode, which you can trust for most low current, small signal applications. This is not the case where you should worry about forward voltage drop.