No air blowing (heat or cold) after driving a while

I have the exact same make, model, and year, and a very similar problem.

Here are the symptoms I’ve noticed:

  1. Happens when the car has been driven for more than half an hour. (takes about 60 minutes for me)
  2. Seems to happen just with AC (for me).
  3. You can still hear the fan blowing inside the dash, but almost no air seems to make it to the vents.
  4. Turning off the air for a few minutes helps a little bit, but the problem returns after only a few minutes of the air being back on.
  5. (possibly related) Ice forms on parts of the AC system inside the engine compartment. LARGE pools of melt water will appear under the car after it has been parked long enough after this problem has occurred.

So, it isn’t the blower speed control assembly, since the blower can be heard laboring away as best it can. As a rank amateur, I’m not sure what the “air blend door system” is, but does this sound like it might be related? And, does the ice sound weird, or is it just me?