If there are lines coming from the transmission to the radiator, it has a cooler in the radiator. If there are lines from the transmission to an external heat exchanger, that would be an auxiliary cooler.
Consider replacing your radiator cap . .
If the smoke is coming from the electric fan itself, then it needs replacing. It could be the smoke is steam that the fan is drawing from the radiator, and you have a coolant leak at the radiator.
The transmission cooler lines are located coming from the radiator on the bottom. One is for the sending and the other for return. The lines are high pressure, just as your radiator hoses are when the car is running. The lines go from the radiator to the top of the transmission.
Whatever seal the fluid is coming out.
The right side (passenger side) would be the transmission cooler lines. The plastic radiator tank is probably cracked near or at that fitting. Replacing the 12 year old radiator would be best.
Pressure and return lines that goto radiator and or transmission cooler to keep the ATF chilled off.
Follow the hose coming out of the top of the radiator; it feeds the water pump.
Because the transmission is in limp mode, and needs servicing ASAP or is broken and needs replacing.
If you are speaking about the two small hoses, yes. That would be transmission fluid.
that is the transmission return cooler line
You may have air trapped in cooling system, try replacing the radiator cap and make sure coolant reservoir is full.