I corrected your pictures to try and help others view them without throwing their necks out or having to rotate their computer screens.
First picture, the foam could have been placed there by the P.O. to force more air to to the other vents to improve the airflow to those vents. That would be my first guess.
Second picture looks just like a normal setup except the Factory A/C cars use a formed hard plastic tube rather than the flexible tube you have.
Third picture, it would not surprise me if that flexible duct is supposed to hook up to the tube there just above the floor. I might be wrong as I don't have a NON A/C unit to look at, but that would be my guess. That might be floor heater tube, but that is much bigger and different than the Factory A/C heater vents at the floor.
Fourth picture, the drain definitely should not be inside the car. There will be condensation coming off the Evaporator core when the A/C if operating properly, therefore that water has to drain somewhere. I would bet there is either a hole in the floor or firewall that the hose in question fits into.
My guess at this point is the previous owner got in there to fix the A/C, got confused/frustrated and gave up. Leaving parts disconnected and/or not connected properly. At this point, if it were mine, I would pull the dash at least far enough out of the way to get to the unit behind the dash to verify what is and what isn't connected, blocked off, broken, etc. Otherwise I think you may be chasing your own tail here with the whole situation. Good Luck!
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