I always think it is a bit of a strange thing here that we all do i.e. try and see what is really happening in a situation where we don't know that much and we have never met any of the people involved. So what I am about to say may be way off. But my thoughts are -
You talk about making sure her life is predictable and secure and scold yourself for shouting when she has hurt you. So it sounds like when starting school, she has gone from a safe quiet haven, say in an English spa, to the equivalent of Delhi with lots of noise, not much predictably and a bit of chaos. A reception class will always be slightly chaotic. So she may simply be struggling to adjust to this enormous change to her life.
Refusing to do anything I ask of her.
She sounds like she has been a fairly compliant child up till now. Seeing other children refusing to do things may have made her realise she does not have to be compliant. I cant remember what age I was, but I still vaguely remember the lightbulb moment I had when I realised that my mum couldn't really make me do lots of stuff.
Telling me she wishes she’d never been born. I think this is simply an expression of unhappiness or even tiredness and being overwhelmed. She is too young to really grasp this concept. It will simply be a phrase she has heard. Maybe sadly another classmate whose parent says to the child they wish they had never been born. I can remember as a teenager saying I wished I was dead. But I didn't really, I simply wanted the negative feeling I had in that moment to stop.
Telling me she hates being a child.
I have some sympathy with this. As a child you have very little control over your life. I would ask her why she hates being a child. But if she doesn't know, I would be looking at how to give her some more control over her life through choices. Some children need more of a sense of control than others.
Telling me she’s useless at everything (despite the fact she can already read, write and do difficult maths).
She sounds hard on herself. I would try and take the focus off what she achieves and focus instead on her personal qualities e.g. you are so funny, and on praising effort put in. You want to discourage perfectionist tendencies which are common in well behaved girls. It is not good for mental health.
This evening she has pinched me, splashed me told me I’m stupid and seemed furious with me. I managed to stay calm until she pinched my under arm and a shouted “stop it” (it bloody hurt!) I’ve never shouted at her before and I’m really upset.
She is obviously angry with you. Dont let her hurt you. But try and understand why she is angry with you. Are you giving her enough space to just be? Are you demanding too much from a tired 5-year-old? She may be feeling grumpy after a tiring school day. Try and arrange things so she has some control over what she does, but you make as few demands on her as you can. But one demand I would be clear on is no hurting you. That has to be a firm boundary.
I would also check with the teacher that there is nothing more going on. But it is important to remember at this age that what seem like tiny worries to us, can be enormous to a 5 year old. So if there is anything it isn't necessarily a big thing.