I agree she needs evaluations, completely. It is very hard to get a thorough evaluation of a 5yo though. Her symptoms will change as she gets older, so the diagnosis will change. She probably isn't able to manifest all of the symptoms at this age. I would look for a developmental pediatrician and/or a neuropsychologist. Something is going on.
I do think that she may be doing too much. If she only falls apart for Mom and Dad, then it may be that she keeps it together for everyone else but she comes home and falls apart because she is overly stressed. I know my boys did this. Neither one could handle more than one after school activity and sometimes even that was too much.
The other thing that comes to mind is the opposite of that. How structured is your time at home? Could your little girl really need the structure and routine that comes at school and her other activities? Is the unstructured time at home the deep seated problem? Chances are that to change things would be a big battle, so it would NOT be an easy fix, of course. It can't ever be easy for us parents, can it? But maybe having a written schedule of what you do when, and sticking to it, is something that she needs. If this is the case, it could be a sign that her problems lie on the autistic spectrum. Not a certainty that they do, but a sign. People with autism often need the rigidity of a schedule or they are uncomfortable, especially children. Or so I have been told. My autistic children only ran on THEIR schedule, not mine.
The other thing about having a child on the autistic spectrum is that often there is a reason behind what they want and what they are battling for. It takes really getting into the child's head to find the logic there, but there IS logic there. It may not be an adult's logic, but it is there. Once you find it, you often can work with it.
I STRONGLY recommend that you read The Explosive Child by Ross Greene. It is truly excellent and very helpful when dealing with a child that just doesn't work the same way that other kids do. I do know how hard it is when your child acts like this, and I am sorry you have to cope with it. Be sure that you and your husband take the time to do nice things for each other, for yourselves, and to nurture your marriage. Those things are all very important, too.