Fluid leak appears to be off of transmission

You don't state what make, model or year the vehicle is, that being said...is the oil / fluid brown/ black like oil or is it red like transmission fluid? Some cars today have what they call sealed transmissions. Yes, they have fluid in them but there is no dipstick to check it with. The transmissions plugs that can be removed and you stick your finger in it to get the level, but you have to crawl under the car / truck to do this. Manufacturers like Kia and Toyota, and others feel that since they put in a high mileage long lasting fluid there is no need for the regular Joe to check the fluids on the car, therefore you can pay the bigger $$$ at 50k or 60K miles and let the deal tech do it for you.

Could also be break fluid check the covers on all the fluids to be sure they are snapped on tight

Blue smoke means the engine is burning oil. It could be a vacuum module that is leaking ATF and some of it is getting into the engine

as the oil in a transmission is used to provide gear shifts , lubricate the bearings and gears , cools the transmission then the expected damage would be bearing failure --pump failure and scoring on the shafts and gears. Having said that I would remove the pan and check for metal in the filter and on the bottom of the pan before I had a fit about the possible damage.. It is possible that refilling to the correct level will all that it needs.

Your radiator needs to be replaced. Chances are you have transmission and oil cooler lines going through them and there are mixing inside the radiator.

usally just a small container (like the master cyclinder holds the brake fluid) for standard transmissions if any.

Actually, on the manual tranny its more like a thin grease. There is a plug located on the side of the transmission, When you remove the plug, you should be able to stick your pinky in the hole and get a little grease on your finger. The fill line is to the bottom of the plug.

There is a drain plug on bottom of transmission and you need a lift. Vehicle needs to be lifted, and ran to operating temperature. With vehicle running, level, and at operating temperature, remove drain plug and fluid should steadly leak out. To fill you need a pump to pump fluid in from bottom. Thanks

On your trany u got a plug just like a transfercase on pick ups.. thats how u check the oil level..

did you recently empty the tranni? like fluid or filter change? If not, there is a leak. I've seen the radiator go bad and tranni fluid gets forced into the cooling system. Check cooling system for reddish oily substance. If there is, replace radiator and refill tranni and cooling systems. may be a good idea to do cooling system flush.