Can transmission fluid enter a radiator and what can be done about it?

Possible. The transmission cooler is usually attached to the side of the radiator. If you look on one side of the back of the radiator, you should see two small lines, one near the top and one near the bottom. Those are the transmission fluid lines. If you do have a leak you'll have to take out the radiator and have it fixed or replaced

A bad head gasket will also make your coolant look bad and is a lot bigger pain....

Have you checked the fluid in the transmission for what may look like emulsion (Frothy substance). Or even the oil may have a different color I would check it both cold then hot. If you are leaking at the head ... you may see a lot more moisture from the tail pipe, lost performance and if left un-repaired ... a clogged catalytic converter