I know this is a basic question but I'm a newer player. When you successfully hit with mind control, after you do what you want with the hit character, does the hit character also get an action token?

No. Nothing in Mind Control tells you to give a hit character an action token.

The reason is that you give action tokens for actions, such as MOVE, CLOSE, RANGE, and POWER. Mind Control simply allows the target to move and/or attack. Neither of these are specified as one of these actions. Of course, hypothetically if it did, it would've been written to give these as free actions so they don't get a token.

FREE actions can not take place inside other actions. So, hypothetically, it would be "at no cost".