Without meaning to dismiss the developmental/behavioural issues that are concerning you, I can't help feeling that the terrible sleeping might be the main problem here.
I think that people who haven't experienced it directly don't/can't really understand how much sleep deprivation (and the resultant perma-exhaustion) can utterly poison your life. Our DD has finally progressed, at 3.8, to being an okayish sleeper, but for much of her short little life she was just awful. I remember all too well her waking up 7 times a night, every night, for months and months and months. Then she'd sleep better for a few weeks, then, often for no apparent reason, she'd be back to multiple wakings. Until she could talk well, she'd often scream, shout and protest, sometimes for hours at a time in the night. Until very recently, she'd be up for the day at 5, or 5.30 if we were lucky, even when she'd been up several times in the night. Her tiredness often made her horrible in the daytime, but she'd fight naps tooth and nail. The effect it had on my and DH's relationship, my work, moods, concentration, sociability, temper, ability to drive or indeed get anything done at all was profound, and distressing.
I smiled in recognition at your post 'Co sleeping doesn't work, sleeping next to him on floor doesn't work, going in and soothing him and coming back out doesn't work, ignoring him doesn't work, gro clock doesn't work, soothing nightlight doesn't work, pitch black doesn't work, calming bedtime bath and massage doesn't work' - exactly. We did all these things too, and as for your DS, they made no difference. I think the only thing that did make a difference in the end is time/DD growing up a bit, which is of no use to you - I don't think you can wait for that to happen, especially if you're dealing with the sleep problems on your own. I've been lucky in that DH has shared the nights/getting up almost since day 1 - I really have no idea how I would have coped otherwise, and I am full of admiration that you haven't completely cracked up shouldering the burden alone. You must pester your HV/GP/anyone who might be able to help until you get some help. I'm sure you've considered it, but if you have the funds, could you employ a sleep clinic/consultant? We used Millpond, who while they didn't miraculously fix things, did help to improve them. Or if you post in 'Sleep', I know people will be able to recommend individual consultants.
With regard to the things you mention in your OP, while DD does love jigsaws and drawing, she sounds like your DS in that she gets very easily frustrated and VERY cross with things like towers falling down, she's a terrible loser (but what child isn't at the age of 3?!), and she can sulk and whine for England. But as others have said, I'm sure they (your DC and mine) are both normal - the spectrum of behaviour and development at this age is huge. I think it's just very hard to cope with challenging behaviour when you're knackered (stating the obvious, sorry!) so that to a well-rested parent, a child is 'spirited', while to an exhausted one, they're a difficult PITA. I too have wasted spent much time over the years worrying about whether DD's behaviour is 'normal' (she is stubborn, has a will of iron, fearless, can throw a spectacular tantrum, has been (and occasionally still is) violent, and just generally likes to impose herself on the world) but have come to the conclusion that that's just how she is. I'm hoping that as she grows up and learns to moderate her behaviour, her strong personality will take her far in life (and not just into trouble
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Good luck to you - I know things will get better, and I hope you can find the people who can help you make that happen, and quickly. Keep us posted about how you get on. I will be thinking of you.