I have one set up in the trainer.
As for coaster BRAKES, they don't have the same feel as a fixed gear and are NOT the same thing. Unlike a fixed gear, a coaster brake can coast and uses a threaded driver to push a set of brake shoes against the inside of the hub shell. That action gives about 20 degrees of play between the brake fully engaged and the bike moving again.
Then, there is single speed. That is just what it says, a bike with a single gear. Usually the bike has a freewheel and a set of BRAKES. My fixed gear has a double sided hub with a freewheel on one side and a fixed cog on the other, allowing me to ride either a fixed gear or single speed by simply removing the wheel installing it the other way around. The bike does have a set of dual pivot caliper brakes.
There is one thing all three have in common: you must maintain chain tension with a wrench.