As others noted -- if this is a public school system, you really may have no choice or very little choice; in most public schools, the child's school is determined by where she resides. It is often very difficult to move a child to another school, short of moving your home.
If she is eligible to ride a school bus, and you made a choice not to use the bus but to drive her instead -- In our system at least that would not be enough for the school system to let you move her to another school; you'd likely be told "She has the option to take the bus" and that's all.
You need to find out from the school system (again, assuming this is public school -- that's the big factor you don't mention either way) what the criteria are for allowing a parent to move a child to another school that is not the one required by her place of residence.
Bear in mind, too, that if you move house and it ends up cutting out her father from the equation, as you note, he may have some legal rights to prevent the move, depending entirely on what your current custody agreement with him is like. (A friend's ex-husband threatened to take her to court to stop her when she wanted to move out of state for a new job; she did end up making the move and the ex moved to the same state. But he could have made her life hell over it if he had wanted to.)
I know this doesn't answer the questions about what to evaluate. All the thngs you mention are important, and I would add that you should check whether the school provides adequate music and art instruction (once a month or weekly or twice a week, etc. -- less music and art generally indicates less funding and an emphasis on "we're only about standardized state test scores").