Your problem could be the door, or possibly the cable.
Look under your dash while shifting the switch on your heater control between floor and defrost, once you identify the lever on the heater unit end, 1st make sure the cable is connected at both ends, then note the direction it goes for each setting and and then grab it with two fingers while operating the dash lever and "help" the door lever in both directions while shifting between the two - does the lever move freely in the full direction for the full range of motion coinciding with the lever, it should be somewhat obvious where the problem is after doing this test. If its not obvious disconnect the cable and check for a full range of motion at the door. If something is blocking the door it is easier than you would expect to pull the whole unit out and shake it empty - may as well check out your hot cold - lever and cable function too.
The cold breeze could be from the vent door on the pass side - they don't always close completely, or a crack in the body (they get kicked alot), or a bad seal where the motor goes through the firewall - the foam seal dries out over time, or if someone replaced/removed it at some point it may not have been resealed properly.
All that said it is a YJ, air leaks are standard - hell, snow blows in my face from between the roof and windshield frame on mine