You should be able to start and go with your vehicle without issue. It is, however, a good idea to let the vehicle fully warm up before pushing it too hard. When the vehicle is cold, lubricants aren't flowing through the engine and transmission as quickly or completely, which means there could be added friction and wear imposed on the moving parts.
WRT the transmission cooler. The transmission works better when cool, so cooling down the fluid no matter what temperature it is outside is usually not a bad thing. I mean, if it's cold enough, it could congeal. That wouldn't be good. I don't think we reach those temperatures here in the States (nor in most of the world) so that's probably not an issue. Hot transmission fluid will do more to damage a transmission than cool fluid will. Again, you don't want to go pounding on it before it gets the fluid moving, so take it easy until it gets warmed up.
You have a misconception about the job of the thermostat. It's job is not to keep the engine cool. It's actually there to keep it at a constant (as possible) temperature. It stays closed when the engine is cool to allow it to heat faster. It opens when too hot to allow the coolant to come back down to temperature. If they only wanted to keep the engine cool, they'd not have a thermostat at all. An engine performs better when at temperature than when completely cool.
Bottom line ... don't hammer on it when cold, but it shouldn't cause an issue using it when it's cold outside.