Try talking to Citizens' Advice. There might be some regulations that could help with the BBQ.
How old are the older children? Could you come to an agreement with them (or their mother) that they may come into your garden twice a week to collect their balls and other toys that cross your fence?
By limiting them to twice a week, at times that suits you, say Monday after school and Saturday evening and making it quite clear that you are expecting them to pick up their things without damaging your flowers, trees etc. you may well find that they are more careful about not throwing things at the fence, or across it.
As a retired teacher, I can assure you that children usually are far more reasonable than their parents if you discuss things sensibly with them.
Tell them you have a bad back, so you cannot pick their things up, that you often have to go to hospital of the doctor's for treatment.
Explain to them that rubbish and food must go into their dustbin, not over the fence. Tell them about rats and mice and the diseases they spread.
It takes a village to raise a child, you know. You might just make a difference to these children, it doesn't sound as if their mother cares all that much.