A/C blowing hot and A/C light is on

The fact you went through a puddle, it probably got the A/C belt wet and that made it slip or the compressor clutch slip, please check the belt. Also, if you let the system cool a bit it sounds like it starts working again which it telling me you might have too much refrigerant in the system, can I ask if you had the system recharged recently? If not, I would lean towards a temperature blend door actuator, they call it an air mix servo motor. This can guide can help and I have included the location of the actuator in the diagrams below. I would remove the actuator and then work the temperature knob to see if it moves okay.

https://www.2carpros.com/articles/car-air-conditioner-not-working-or-is-weak

Check out the diagrams (below). Let us know what happens and please upload pictures or videos of the problem.

This incident had nothing to do with anything getting wet. That was just a separate example I used to show that when the A/C system stops working for another reason, the light flashes. This time the A/C system stopped working; the light stayed solid. Everything on the same fuse worked fine. When I pulled the fuse and replaced it, the A/C started working again. What could have caused this behavior?

It sounds like you reset the system when you pulled the fuses which can be the air mix door actuator. Also, the system can be low on refrigerant. This guide can help:

https://www.2carpros.com/articles/air-conditioner-how-to-add-freon

Please run down this guide and report back.

Charged the system. Blows cold, but I am still having the problem where the A/C unpredictably cuts out. I don't hear any clicking sound that would indicate a bad actuator. Sometimes if I smack the center of the dashboard, the A/C will pop on. I'd just buy an actuator to give replacement a shot, but they are not cheap; What would you suggest to do next?

This sounds like the blower motor is going bad. To be sure let's check the power and ground to the blower motor once the air stops, this guide can help:

https://www.2carpros.com/articles/blower-fan-motor-works-on-high-speed-only

The blower motor location is tin the images below. Check out the diagrams (below). Please let us know what happens.

There doesn't ever seem to be a problem with air blowing through the vents. The air blows fine, it just doesn't blow cold. That's why your suggestion about the actuator made more sense. Does the A/C have a separate blower? Your thoughts?

No, it doesn't. it could be the HVAC controller if it isn't the blend door actuator. Let me know. Check the images below.