OK, I am back...
As I said, this sounds like classic attention seeking though there may be more going on re. being unable to control behaviour.
So we went through a bit of this where DS couldn't walk past DD (2 1/2 year age gap) without pushing her/pulling her hair and he STILL does silly things (TBH some of them are not that silly and DD overreacts e.g. he growls at her or LOOKS AT HER FUNNY) because she just makes such a satisfying noise when he does.
We got this book:
www.amazon.co.uk/Calmer-Easier-Happier-Parenting-Revolutionary/dp/144472990X?tag=mumsnetforum-21
Get the first chapter free on Kindle but definitely get the whole book too.
The two techniques we used which helped a lot (and we need to remind ourselves to use them again) were Descriptive Praise (the whole crux of the book) and re-dos (which is used in a lot of other techniques too).
And the other thing that we use is the "baskets" or priority system - some things are red (top priority), some are yellow (ignore if possible) and some green (always ignore).
You can't deal with everything at once!
So:
Every time he's even NEAR his sister and not hurting her, you tell him how good he's being and describe what he's doing. He's sitting nicely, he's keeping his hands to himself, well done.
If he does push past/hurt her/throw something to get attention, ask him to try again to walk past her/how else can he get her to look in a nice way. Praise him for doing it properly (and don't tell him off/give him attention for doing it wrong).
Other priorities:
Get a table and chairs so you can all sit together. Probably not a priority to make him sit down, but praise him when he does. Maybe this can go in yellow.
Bedrooms - can he go on his own for the moment, and your middle child come in with you if there's no extra space? He is getting attention from her when he wakes her up (and probably from you when you tell him off). So waking anyone up just gets a boring "back in your room, it's not time to get up".
Might be worth getting a play pen for your middle child, so she can escape/he can play with one of you.
He needs 1:1 (all your children do) - you can spend time with the younger ones while he's in nursery thankfully - and he may have to have it at the weekend but there's also story time after the younger ones are in bed etc.
Breaking things that are his - his problem - just ignore and say "oh dear".
Same with picking at the wall etc. Nobody is getting hurt and you have other things to deal with.
We do use a form of time out but not with DS on his own (he'd never stay where we put him for a start!), he is in his room and we are either there being boring or just outside. We ONLY do this is he is being unsafe (throwing/hitting) and it's actually only recent; we saw improvements with the other behaviours before this and it's only as he's got a bit older and more "aware" of what he can do to get attention, that we've had to use it.