I donèt know what to suggest, because I am in the EXACT same situation, it is almost as If i was writing...lol just wanted to let oyu know oyu are not alone!
You know, for some children, 3 is just too young to build out social skills. Some children can, and some can't. For day care, any chance of moving him to a sitters house or place where there are fewer children, so he does not have to navigate this arena until he is older? Of course, he has to live with his brother, but does he HAVE to play with him? Can he just be left alone to play for most of the time?
Also-- I totally get the sleep thing-- not enough sleep, can't control himself. Gets enough sleep, big difference. I got one of those myself!
If you really believe that he has some impaired social skills-- then I suggest talking to his dr about a referral to a specialist who could evaluate him and give your daughter some suggestions for encouraging positive social interactions and discouraging negative social interactions.
Also, this is just my opinion-- but trying to teach a child not to be physically agressive by using physical punishment is generally ineffective. I'm not a anti-spanking nut, per se-- but I have found that children who have aggression problems are not effectively disciplined in this manner-- hitting is their "problem," so its not really working to use it as a "solution."