Can I put a window air conditioner in the doorway?

On the floor is not a good idea.

Heat rises, remember?

You would have issues with blocking your fire exit by blocking the doorway.

If the heavy drapes happen to fall and block any of the air vents on the sides or back, it can overheat and cause a fire.

If you cannot use a window, you can buy a portable A/C (on wheels), close the door, and have some cool air.

They still should be vented outside though. A dual vent system is more efficient (draws outside air in to cool the condenser and blows out the hot air and moisture back outside through a window)

The AC also exhausts a large amount of moisture which will accumulate and cause mold.

AC doe not make cold.  It moves heat from one place to another.  Window a/c (actually all A/C)  has two fans.  One recirculates inside air and one circulates outside air.  Don't let the air from one mix with the other and it'll be OK with the curtain you're talking about.  BUT as others have said, there's ALMOST always water condensed that you have to contend with.  

There is no exhaust. The fan only works for the front pulling warm air in through the filter and blowing out the top. The back is just coils where the heat is dissipated. Find some kind of tray that you can set it in to mop up the condensation it creates to not ruin the floor, and use something like a sleeping bag to isolate the front from the back in the door frame. Will work fine.

NO.  You will destroy the floor.  Window air conditioners also give off moisture which usually drains OUTSIDE from the back of the unit.  (one year, during a drought  in my area, I placed a five gallon bucket under the drainage area of my window air conditioner.  I caught enough water from the drainage of the air unit to be able to keep my flowers watered and save my decorative bushes without actually using water from a faucet - we was under a "do not water lawns" order.)

So - it isn't just the "exhaust" from the unit - it is also the CONDENSATION that you need to deal with.  

If you want to place an air unit in the door way to the room - you would need to brace it in the TOP of the door frame and place something under it to catch the condensation.  You could then walk under the unit instead of stepping over it to get into the room.

I've done that before without issue.