Also, when you live in suburbia and have a trampoline it is expected that you may see children bouncing on trampolines. it is good exercise and good fun and people need to show some tolerance.

Where we live any overgrown bushes on our side of the fence are our responsibility to trim. Maybe grow some plants that cats do not like as a deterrent. Maybe go to the nursery for recommendations for some plants. People I know swear by plants to keep cats away.

I’m pretty sure if you want the fence redone they have to pay half, then if you want it higher than normal and they don’t then you have to pay extra.
As for the cat. Most councils hire out cat cages that let you catch them in your yard and take them to the pound. I know cats around here have to be kept in your own yard and owners have to pay to let them out.
Not sure about the tree. You are allowed to trim anything hanging over your side but its up to you to get rid of anything you cut off. Keep it trimmed regularly and it should fit in the bin.
Otherwise its all their problem. They live next to kids. Their bad luck. You offered a solution and they said no.

I would call your local council and get some advice from them regarding the tree.. As far as I know it definitely should be their responsibility to keep it trimmed. Then you can lodge it with them to send them a letter, or try to resolve it yourself.. Good luck!