Using 2n2222 to switch relay on/off with 3.3v

Yes, H stands for 'HIGH' or 3.3V and LOW stands for 0V.

No, It's a resistor that limits the current to the gate of the MOSFET. you can also use commonly available 220 ohm value. But that would increase the switching time.

What you have circled is the MOSFET itself. The BJT takes a 3.3V logic input, inverts it and then amplifies it into a 12V signal for the MOSFET to turn on. The MOSFET is switching the 1K resistor (load).

I use 10K ohm as a standard value for pull up/down resistor.

@neketege88 I created this circuit for simulation for your understanding. This circuit is for understanding purpose only, not suitable for your relay. I made this for SONGLE SRD-05 12VDC Relay. There is a small voltage drop (VDS) of around 200mV of the MOSFET so the mentioned relay also might not work in some cases.