"On my mind" is used for something you're thinking about, like "I didn't sleep well because my exams were on my mind." "In my mind" is usually used for something that's imaginary or an illusion, like "I thought I saw a ghost, but it was all in my mind."
I'm not sure how you mean to use it, but the phrase is usually "in the background."
Yes, that is OK for 'in her mind'. As for your 'background' phrases: again, either is possible. When posting to forums, please supply complete sentences for discussion.