Can your character be affected by mind control? I'm really conflicted on this; Zatanna was able to completely boot J'onn out of her mind even while being unconscious and affected by outside sources, but she was shown as having forgotten Max. Presumably, if he focused on just her and put a lot of effort into it, he could! Once she's aware of something like this having happened, though, she'll take necessary precautions to shield her mind.
Is there anything he just CAN'T make them do? Getting her to hurt or betray her friends would be difficult, depending on the extremity.
Are you open to him pushing your character's mind? Sure!
If so, in what ways would they be most vulnerable to manipulation? She's most vulnerable when it comes to her relationships with other people. Like most mages in the DCU, she has a track record of all the people she becomes particularly close to dying or using her. Plant the seed, the rest will follow.
Are you open to him erasing/implanting memories (if so, what kind?)? Maybe! If it's toward a specific end, I can assume some sort of fluke in her mental defenses.
Is there anything you don't want him pushing for/mentioning ever? Nah, there's nothing sacred!
Are you open to him pushing other characters for information on your character? Knock yourself out.

TL;DR. :(