She messes up Robots bad.
That's by far her best feature.

Decent dial for Shadowcat that we needed.

If you moved her back and forth through characters with the keyword, would it deal damage multiple times?

No, why would you think it would?

If you could somehow give her Sidestep, you could trigger her power multiple times per turn with that and her regular move action.

No you couldn't, sidestep is a free action, the power says "If during a move action she passes through a square containing an opposing character" since you can never take more than 1 move action a turn there is no way to cheat this power.

Well you are partially correct, sidestep is not a move action but there are a few ways to give a character more than one move action per turn. For example GCPD ATA.

WXM shadow king can mind control her and then make her take two actions, one of the actions must be a move action. Why can't the other action also be a move action? They would be two separate actions.

Yes, but Shadow King can't target friendly characters with Mind Control. So...you'd be targeting an opposing Phantom Girl. Meaning you'd be helping her wreck your own team. Because she only messes up opposing Robots/Vehicles. So, umm...yeah. Not exactly effective.

You mind control Phantom Girl she becomes friendly to you.

This is no different than just having her attack a character.
The vehicles and robots that are on your opponent's team would be opposing characters to Phantom Girl at this time.

Oh. I guess I overthought that and got stuck on trying to squeeze extra move actions out of your own character.