IMO there is a limit to what kind of behaviour you can and should ignore. A three year old should not be allowed to hit or bite anyone, but told that hitting or biting hurts and is not allowed.
He is still small enough to be dressed by an adult, if he refuses to get dressed. Say something like, "Let's pretend you are a baby, and I will dress you." If he protests, "So, you're a big boy then, so let's see how fast you can get dressed by yourself."
The boy is confused by the split between his parents and probably has gathered that mummy isn't well, so it's no real wonder he is being difficult.
I once had a six year old pupil who turned to me and said, "It's nice you tell us what to do and why you would like us to do it and it's always the same rules. Could you tell my mummy and daddy how to do that?"
In other words, please behave like grown-ups and make it clear to the little ones that some things, like getting dressed, that just have to be done.
If he is awake and full of life at 5.30 what time is he going to bed? If it suits to keep up the afternoon nap, then let him stay up a bit longer at night and see whether he might then at least sleep until 6 a.m.