C.D.
Boy, that must be frustrating!! My 4 year old does not take naps anymore. He goes to bed at 9 (I wish it were earlier) and gets up at 7. He acts fine during the day, so I guess he doesn't need any more sleep. He does get up in the middle of the night and come to our room wanting daddy to sleep with him. That is a bad habit that I wish my husband would break. (His nightly routine involves him staying with my son until he falls asleep- that's the issue) Anyway, I brought that up because maybe he doesn't like sleeping in his bed by himself and that is really the issue. That is why he is getting up at 2, then he probably stays awake as long as he can, then falls asleep again for one more cycle then wakes up at 5:30. So, maybe make sure his room has nothing "scary" (ask him) and then make sure he has a lot of stuffed animals ("his friends") to help him if he feels like he needs a friend at night. Make sure he has a nightlight in his room, maybe the dark scares him. Also, make sure you have room darkening blinds. In the summer, it gets like out about 5-5:30. Maybe he sees that it is light outside and it is time to get up. JcPenney (online only) makes wonderful room darkening blinds (the best I have found). I think the brand is "Carmel" or something like that. Check out their website. Finally, if he does get up at 5:30 take him back to his room and tell him he has to go back to bed and that it isn't morning yet. It may take a couple of weeks for him to give in and listen and not come back to your room, but consistency is key with this age. Every time he comes in your room you take him back to his room. A friend of mine bought a wall clock and put a star on the hour hand and colored the area between 7AM and 9AM with a blue marker. She told her son he could not leave his room until the star was over the blue area. Reward him somehow if he gets up after 7. (Reward chart with stickers and prizes works good for us) During the day, make sure you are giving him a lot of personal attention (one on one is best, maybe while sister is sleeping you can lay with him in his bed and read books, or watch a movie) and praise him every time he is doing something good. He will want to please you more if you do this, and maybe it will help speed along his nighttime issues.
As for his hitting his wall and waking up his sister, we have a air purifier that we have set up right outside our son's room that drowns out any noise. Maybe you can put one of those in her room to help drown out the noise from him banging on the wall. Also, make sure you discipline him EVERY time he bangs on the wall. I know you will be too tired in the middle of the night, but he will hopefully catch on quickly.
Hopefully your doctor can give you some good tips as well. Good luck!