Does using MoveTo on the player (to move to a different cell) automatically move companions to him/her as well?

Nope. Unless it's done on the companions itself. But if it only says player.moveto xxx...no.

Hmm I was afraid of that. Thanks for the reply.

A lot will depend on the companion's script. It's possible for the a companion to have an object script that essentially says "if I'm more than xxx distance from the player, MoveTo Player" to make sure they don't wander independently. If that's the case, then it doesn't matter how the player gets to another location, the companion's script will (once the script runs) compensate for that and catch up. It all depends on what mod is providing the companion, and how it was scripted to behave.

And usually companions have a script like that, otherwise they would often get lost. So you shouldn't worry.