Drama - you ARE strong. But you are under a hell of a lot of stress. I'm going to break your list of 'to do' things/observations down...
No chance of leave. Unless I get signed off by the doctors. School are being awful. Get signed off. The kids will miss you but the adults are being shits. You need some space. By all means note down what you need to do on a daily basis for your job, what meetings are coming up, but you can then leave that aside knowing that you've covered everything from your side.
(I remember being off sick from work and my boss rang me every day "to see how you're doing". But really to find out when I was coming back. I got the doc to sign me off for two more weeks with stress. It really helped me calm down and reassess).
I should have prevented this and have let everyone down.. No you have not let everyone down and there is no way you could have prevented this. You were not to know that some people in your family are back-stabbing, lying bastards to such a degree. Any decent person would expect others to be decent.
The stuff going on about school makes me feel like I am crap at my job. You can't keep all the plates spinning all the time. You're not crap - you need a bit of space and healing time.
Dd regressing with lots of wet pants, continually wanting her dummy and refusing to sleep more than an arms length away from me makes me feel like a crap mum. - This is how her grief is manifesting itself. She wants the security of having you nearby - someone she loved has effectively 'disappeared' so she's going to be clingy. You're doing all the right things with the books. If you need to cuddle up together and cry, do it!
The state of my house make me feel like a lazy git. - Housework is the devil's arse and can be ignored. As long as you have a bed to sleep in and a plate to eat off/something to eat with, it's fine. If it bothers you that much, rope in some mates to have a 'cleaning party'. You supply tea/cake. They hoover, wash up, etc.
The fact that I can't find paperwork to enable me to trade in and collect my new car makes me feel like an idiot. - You're already trying to cope with a hell of a lot, it is no wonder you're mislaying documents. I did this when my Dad died. Mislaid all the documentation relating to probate. I was trying to cope with DH working 6 days a week, rest of family not dealing with Dad's passing, and college. Got there in the end, but I had to let some things sit unattended whilst I worked through my own grief.
Dh having to work his usual long hours makes me think I won't cope with 2 children when I can barely cope with one and am doing so many things wrong with her anyway. Is there any chance DH can get some leave? Or a friend can be on hand for a few hours a week? Nobody is a perfect parent. As long as she's clothed, fed, cuddled and loved, you're doing it right.
It has dawned on me I'm not talking about this to anyone in case they agree that I really am an all round rubbish, lazy person. Talk to your GP. Talk to your friends. Talk to us here. You are absolutely not rubbish, nor are you lazy. But you are trying to be all things to all people and putting yourself so far down the list of 'people to care for' that you're falling off the list.
I want to be on top of everything but have no idea what to tackle first. You've handed over stuff to the Sol, and to the Police. Keep records of that, and consider those things out of your hands. MrsCake has a great idea about the memory book and the photographs.
My confidence has really been shattered by all this. Of course it has, love. You'll get it back, I promise. However, it's not an overnight fix. Step by step, things WILL improve. Right now, you're at a low point, quite understandably.
We all like to think of ourselves as copers, as strong, as invincible. It takes it out of us to discover that we're not. When we're constantly being undermined, and taken advantage of, then confidence will be lost.
Any one of us who is near to you and can help will do so willingly.
Right now, I want you to get yourself a 
. I want you to take a really deep breath. I want you to give yourself a huge hug. Now phone the doctor and make that appointment.
If you want to offload, please feel free to PM me.