Is my circuit to power 38 kHz LED matrix correct?
Thanks for asking the question. Let's discuss step by step,
1. Is the MOSFET section of the schematic correct? (what can I read to understand how to design these?)
No, it's not. You have to connect Mosfet Source to 5V directly and You have to use a gate discharge resistor(10K) between Mosfet Gate and Mosfet Source. Use Gate resistance of 100Ω for switching at 38KHz.
2. How do I choose a value for the resistor next to the MOSFETs?
It totally depends upon how much current your gate driver can provide and how fast you want to switch MOSFET. In your case ON STATE RESISTANCE of Mux us around 100 ohms and ESP32 can provide at max 40mA@3.3V, so you can use any resistance between 82.5Ω to a higher value. 100Ω will give you a decent switching speed.
3. Do I need a capacitor somewhere?
Yes, You have to use a 1uF ceramic disc decoupling capacitor between Vcc and Vee of your Mux and You have to use 100uF Capacitor between 5V and Ground near to LED Matrix.