chelsy I think there are essentially two issues here, the problem of you being denied some of your employment rights, the impact that could have on your now and in the future, and the more immediate problem of what you do about the time you are having off.
Lets think about the childcare problem first. You are right of course, not everyone has their mum or whoever up the road, lots of people don't. But that's obviously not the employer's problem, so the issue is what other emergency arrangements could you make? You need to have an option in place, something that you could do in this situation. I understand you need the money from this job, but if you are having to take unpaid time off at the moment, it wouldn't really make much difference if you had to pay for this emergency childcare either. You need to have a proper think about what else you could do. Are there any neighbours, any students, does anyone know anyone who would be willing to look after them once in a while for some cash? What do the other mothers locally do? Try asking them at the school gates. There are such things as emergency nannies, which would obviously be expensive, but if it saves you taking unpaid leave and keeps you in a job, might be worth considering. Try putting a card in the local shop window or advertising on gumtree for ad hoc emergency help. Sometimes people like this advertise their services on gumtree as well.
As I say frequently, you need to work out what a positive realistic outcome would be to this and take some action towards getting there. You know that keeping taking unpaid leave like this isn't a realistic long term solution, it sounds as though your employer has actually been very understanding so far as it is. So you need to find yourself a better solution going forward.
I don't obviously know what the illnesses have been, but if it's been very frequent, is it worth a trip to the GP to have a chat about whether there could be anything physical or otherwise wrong on a longer-term basis that needs looking into?
I can't give you a magic solution to the childcare problem because there isn't one, but you need to find something. Try asking anyone locally you can think of, local NCT might know of someone that does this for example. I know where I live there is a directory of childcare on the district council website, try there maybe? Try asking on the CM/aupair/nannies topic on MN to see if anyone there knows of emergency childcare providers in your area.
In terms of your contract and the other issues around that, I find this strange. It doesn't sound as though your employer is a bit dodgy and deliberately trying to get away with doing anything nasty, he sounds helpful and flexible and as though he does value you. I think it's more likely he is ignorant of his responsibilities. If you are paid by the shopping centre I would expect them to know this and address it as well. It may be that they are technically your employer, not sure. Are you paid into your bank and do you get payslips or anything, or is it cash in hand? I would anticipate the shopping centre probably pays a few people so they should have something set up. Have you got nothing in writing at all, not a letter or anything, and who did you give your bank details to if you're paid into your bank?
I think the childcare is the important issue to sort out, as if you keep taking lots of time off the other stuff is likely to become irrelevant if your boss gets fed up with it. But I do think the employment stuff needs getting to the bottom of longer term. Have you spoken to anyone else who works there about their arrangements? It's not such an immediate concern now, but what if you (for example) got pregnant again, what about your maternity rights, that's important. There are other similarly crucial rights that could be an issue down the line.