Sounds like grandma has encouraged a bad dynamic with the kids. I assume she watched the older son too? If she reacts to stress physically, what an example to set.
I think it's a good sign that he is well-behaved in school. At least that says he is able to control his impulses for part of the day. I have found with both my kids that six is a really good age for behavior charts, penny jars, those types of incentives, where a child earns something for good behavior during the day or can lose some reward if he does not behave well. I think it helps kids to begin to develop their internal ability to control their behavior and plan their actions (with small, manageable tasks like "brush teeth" or "make bed").
I'm not sure what to advise about the aggressive behavior - it's luckily not something I've had to deal with much. But 6 is a volatile age for some kids and having more consistent structure around him might help in general with moderating his emotional state. Also, they should make sure he gets enough sleep.
Not sure what to say about grandma since we don't have a lot of information - but it sounds like she also lacks impulse-control, so a more structured daycare environment might be a better choice for the younger child.
Oh - and buy her a copy of 1-2-3 Magic. I didn't follow it completely, but I found it very helpful for about ages 3-6. And it's simple to follow.