it sounds to me that you are doing a wonderful job in extremely difficult circumstances. I understand only too well why you feel things can't go on like this much longer.
You have already received some very sound and helpful advice. I only have a little about the two children.
You say they are doing very well at school, which is great. All the same I think you should get in touch with their form mistress (sorry I don't know the current term) and explain that you are worried about the way they behave at home, and when you take them to play with other children. As they are obviously well behaved at school their teachers may be able to help by telling you what works at school. (No child is ever perfectly behaved all the time at school, believe me, I used to teach).
The next thing is that you will need to try explaining some ground rules at home, or simply demanding better behaviour from them in your home. Along the lines already suggested: we don't behave like that here. The eight year old is certainly old enough to understand some firm rules that are adhered to. If or when he oversteps them, tell him that is not the way to behave, that he is not allowed to hit out, or whatever it is and sit him down on a chair, or stand him in the corner, with his hands behind his back (my mother's favourite punishment for naughty children). If he moves off, fetch him back, telling him that he is to stay there until he has counted to 100, or until a certain number of minutes are up by the clock, or until the egg timer rings, whatever you find suitable. The younger one should be able to understand these rules too, although it may take longer with the little one.
I think the rules must apply to both children - there was the same age gap between my sister and me, and I, as the elder, was dreadfully angry when she got away with behaviour that I was told off for. As grown ups we know that some bad behaviour has to be excused the younger child, but explain to the elder that the little one is too small to know it is wrong, making a point of the fact that the big boy knows how to behave. Tell him that when he behaves nicely he is helping the little one to learn what to do, as he of course looks up to his big brother. (Flattery will get you anywhere!)
It seems as if your daughter has got the message across that their bad behaviour is hurting you, but this may just be momentarily. If it is, remind them that they have been told they are hurting you. Reassure them that you love them and that when you get cross you are cross about the way they are acting, but that makes no difference to the fact that you love them, just as much as their mother does.
It is difficult explaining divorce to children who would all much rather have mummy and daddy living together. I would make sure the children realize that mummy and daddy have decided not to live in the same house any more, but that they still love their children. Unfortunately, you probably won't get away with not telling the older child why mummy and daddy aren't together any more. Sooner or later, he will demand an explanation. You hint daddy may already have provided one, so perhaps you should say as calmly as possible, if your daughter agrees, that grown ups sometimes just don't get along anymore, even although they love each other, and when that happens it is best for everyone if they don't live together.