It sounds like both of you children need some non aggressive redirection. Set up two places in your home for this. One is a time out chair, where they can hear what's going on but can't see and the other is a wall or door where they can put their noses on. When daughter starts to push brothers buttons, she has a choice on one or the other and there she must stay without talking or turning around for about 8 minutes (because of her age, the older she gets increase the time by 1 minute)and the same goes for your son, but he gets only 5 minutes. If he disrespects someone or something, either the chair or the door/wall. Now for him not picking his things up, give him a choice, either he picks up or he'll loose his belongings. Put toys, books, clothes into a box and put them on the top shelf of your closet or somewhere he can't get to them. When he's lost all of his toys, explain to him that they are tired of being disrespected and abused and that when he learns to respect his belongings then he'll get them back. Don't give in either. Now for his throwing of temper tantrums, send him to his room and tell him that he can throw them in there because you don't want to hear them anymore and when he's through, then he can come out with the rest of the family. These are some of the things that you can do at home (time tested in this house and they do work).
Now if your son is going to a preschool of some kind, have them test him for things like ADD and ADHD. The reason that I say this is because I have two sons who's behavior was about this same way and after testing and getting them to a psychirtrist, who reviewed the results, said that they were ADHA and ADD respectfully (older son is ADHD).
There is no quick fix and no easy way of doing it either, but you need to do something before you need a very long vacation away from both. Not only have they learned to push each others buttons, but they're starting to learn how to push yours as well.