First, my children would not be allowed back to grandma's house until I figured out why this was happening. Grandma would be welcome to come to my house and spend time, but Grandma has to be the one to leave, while child is screaming and crying about her not wanting to go. I would not accept always being the "bad guy."
Also, I would want to know more about what happens at Grandma's. Somethng about the attention, the environment, the activities -- something is happening there that she really likes.
The chewing sounds like it may be a sensory issue, so I would go to an Occupational therapyst before going to a psych. The OT info would be helpful to the psych.
Regarding the hitting -- what is happening before she strikes the other child. Is she being teased, picked on? There are reasons for these things.
Also, if she is chewing of a reasons, and the teacher is scolding her infront of everyone, then the other children learn that picking on her for chewing is acceptable because that is what the teacher is doing -- basically.
My daughters' Pre-K has a child with a chewing issue (I am not sure which one, they did such a nice job handling it.). Because of this child, all the children were given a "chew necklace" that they made and washed as part of the regular day time routine. Anyone could wear their necklace and chew on it whenever they wanted. It was great, because chewing is helpful to calm kids down, regardless of their issue.
Another thing to consider, is that there is nothing wrong with your child, she is just in such a stressful place right now she isn't able to cope and behave as you would like. Maybe she isn't ready for school, or maybe this isnt the right school environment for her. My daughter had terrible problems in school in first grade. They had us completely convinced that our daughter has some real serious problems, finally we switched her schools. The next three years were a dream! When I asked her second and third grade teachers if they saw any of the problems the other school had mentioned, they looked at me like I was crazy. Alos, the problems we noticed at home went away because our daughter was just generally happier being in a school, where people liked her, for who she is.
Makes sense when you think about it. Most important love her for who she is, and help her. Something not right is happening and she is 4, she cannot tell you, you have to go find out! Good luck detective Mom. :-)