Does it matter which side of radiator the transmission lines hook up to?

You generally want to hook them to the transmission side of the radiator. The engine side works ok if you can get the hoses to fit but then your engine doesn't have anything to easily connect to.

Beyond that, it's in one end and out the other. Like Thai food. If you're running an auxiliary cooler you can run them in series or omit the radiator cooler as long as the aux cooler is big enough.