Does something at school cause him to be anxious, does the anxiety cause the aggression to come out? There's a reason behind the behaviors... I would investigate more what is happening at the school. You may never fully know. what precipiated him hitting the teacher and the cart?
If he hit the cart the teacher was pushing, I am guessing it was at recess? It sounds like his problems happen more on the field than in the classroom? Not unusual, if true. Play time is typically unstructured. Kids will do things to kids that get them going. Some have more control, others don't.
Frankly, I need more info, and maybe you do too. Hitting the cart sounds bad, but I can think of many reasons how that could have happened. But I can tell you this. I have seen one to ones work great. The para kind of stands back and watches. Only gets involved if things get carried away. A good para can be almost invisible. But it takes a while for the child and para to develop that kind of a relationship. I don't like homeschooling (unless its the last option), because the child loses out on contact with other kids - and while it creates problems, its extremely important for their development.
They need to learn the social interaction.
And yes, I would try and find out more about when he is having problems in school. It maybe that its happening in an area that can be controlled better. The other thought is that you need to model at home proper behavior for him. If he is getting to physical with his brother - you must immediately stop him. Explain why. Let him start over. Don't punish - teach. If you practice this at home, it will carry over to school. If you don't, its tough for him to know the boundaries. good luck!