2001 Audi A4 heater blows cool air even when set on Hi. When I park water drips from AC condenser like I was using air conditioning

Problem with driver's side blend air door. Water on the floor is a sign of a clogged condensate drain tube. Do a search for how to clean a car condensate tube.

The heat exhanger in the ductwork will create condensation as it cools the air, removing water from the air. There is a rubber tube that can become plugged with acumulated dust and dirt in the ductwork. That tube allows condensation to drip onto the ground beneath the firewall in the engine compartment. Try blowing compressed air into the drain tube from beneath the vehicle. A blocked drain tube can cause mildew to grow in the ductwork, which is a bio hazard.

A properly functioning condensation drain tube will small water puddles to appear under the vehcile where it is parked after the AC has been used.

take the top off the filler cap run engine to boil. keep it full off water to the top. you will see air bubbles coming out as the engine gets hotter. squeeze the pipes on the rads to help the air come out. when fan kicks in turn engine off and leave to cool down.while you are doing this make sure the fan for heater is off but is on full heat. can have fan on number one on speed that is fine.

water dripping from a.c after being used is normal on any vehicle, its just condensation frome warm air and cold ac lines, as for rain its probably just rain water coming through the front end, however if the water is reddish or green in color it would indicate a coolant leak and would be noticable much more often.

If anti-freeze is actually coming out of the drain tube then yes, you could very well have a bad heater core. You may also be able to smell antifreeze while inside the vehicle if it is leaking. [Its a kind of sickly sweet smell] If just clear water is dripping out then this is most likely from normal condensation that is collecting and dripping off of the air conditioning evaporator core as the air conditioner runs. This condensation collects in the heater, a/c housing then drips out of the same little drain. Water from normal condensation will evaporate off of cement or pavement pretty quickly but antifreeze takes quite a while to evaporate away and feels slimy to the touch. It doesn't take much of an antifreeze leak before the cooling system gets low on so make sure you don't run low on coolant which can cause the engine to overheat.

heater core clogged or heater valve

You likely have a restriction in the heater coolant hose circuit - perhaps a coolant control valve not fully opening, or a kinked hose.
Insufficient coolant volume is flowing to keep the heater core hot.
Try back flushing the heater core system with a garden hose to check flow.

Recharge AC. You can buy a recharge kit at Walmart, or any parts store, and do it yourself.

yes, very possible. but first check your fluid level, thermostat.

your heater core went bad, you need to replace the heater core....