Your statement that "no air comes out at all" to me indicates that the blower/fan is not working. This could be a problem with the blower switch/speed control, the blower motor, the associated wiring, or a fuse which protects all of the previous.
I would remove the blower (often accessible from the engine compartment, on or near the firewall) and test it to see if it is operating. if not, you would need to troubleshoot to find the fault. If it is working then you would investigate for a possible blockage, perhaps an issue with the temperature blending gate/door.
The fact that you "sometimes... feel a tiny bit of heat at the vents" could just be air blowing through the ducting and across the heat exchanger, due to driving speed.