Where should you film the audition videos?
Against a blank wall, preferably white or grey (beige will work, but can give a slight yellow cast to the video, not a huge deal, but not as good as white or grey). Other backdrops can work, but a blank wall removes the possibility of distraction. Good natural light is preferable, but the only fundamental requirement is no shadows on your face. Generally speaking, the more light, the better (but not so much it washes out the video). A school classroom or church classroom will work.
Should you have a live accompanist?
It is harder to balance the sound if you use a live accompanist - the piano can easily overpower a vocalist in a recording if not carefully balanced). You can use a recorded track unless a school specifically says you must use a live accompanist (don’t know of any that require this anymore).
Should the dance portion be in a studio?
No. You can use any room with good light. A school classroom or church with the chairs pushed out of the way will work. We used a room at a church with the floor cleared.
Do you have to make a separate one for each school?
Not necessarily, but there are differing requirements between various schools. Some want 16 bar cuts, some want 32. Some want a Contemporary song written since 2000, some want a Golden Age ballad with a sustained legato line. Some want one monologue, some want two. Some schools want an introductory statement about why you want to go there. You have to go research each school to see what they want - most of them are on GetAcceptd.com
You can do a perfectly acceptable video with an iPhone clamped to something stable.
Here are some threads to read:
http://talk.collegeconfidential.com/musical-theater-major/1795731-videos-p1.html
http://talk.collegeconfidential.com/musical-theater-major/1795731-videos-p1.html