I have not had the extreme experiences that you seem to be going through, but my son used to have a lot of temper tantrums and outbursts.
A few years back we did a behavioural therapy through the BU center for anxiety in Boston.
The big thing I carried away is to give your child 5 minutes a day, where she can pick which activity you and her together will do. My son started looking forward to these few minutes (which usually turned into 15 or more) and he would plan what he would do. It could be having a story read, doing some drawing, playing a game... just remember, the kid is in charge, and if she wants to draw something impossible, not to stop her, but support her. It worked quite well, and really gave him a big boost to his self esteem and we spent quality time with each other. It also had another benefit: he actually started to do some of the activities on his own at other times of the day. He was a lot happier to be at home and could focus on activities.
I find it normal if a kid is loosing it at home after a long day at school. If she is fine there, it seems to me, that she can cope to some extend with the pressure of school, but at home, she'll need to get rid of that tension. If she has anxiety issues and is in school/afterschool all day, she has to be exhausted at the end!
Now, sounds like your girl has a bunch more things going on, and I think you are right to figure out what it is.