You can't do much about the different height of the fences. My garden slopes at all different angles in different parts of the plot and is wedge shaped .
Forgive me, but it sounds like the worst idea I've heard of for a garden of that size ,, ever.
Shame on you. I do understand your problem with kids though. Have you thought of Astroturf - you know the stuff they use for soccer pitches? What you can then do as the kids get older and don't wreck everything, plant shrubs round the side... Good luck. Ian
I had a patio and path laid last year, but I was very surprised how slippy the slabs were during the Winter, felt most unsafe walking on them.
Chilldren just enjoy running about for the sake of it and having grass is so much safer.
You can put up a tent, or roll on it and play football. It's softer to sit on, jump on and all the stuff small children do. Like tripping over their feet or falling off climbing frames and trampolines.
For the job to be done properly you are looking at well over £100 per square metre, probably nearer double that. It is a major undertaking. From the photo it looks like something in excess of 100 square metres so you do the maths, preferably when sitting down.
As others have said, paving does require maintenance too, and if it doesn't get much sun can become slippery from algae and moss. That needs to be cleaned annually and requires a decent pressure washer to do it.
Looking at the current state of the lawn I can certainly see why you're considering removing it, but it would be cheaper to get somebody to come in and look at ways of recovering it. A local garden centre or nursery should be able to put you onto somebody who can help.
How about using porous paving such as gravel which will allow the rainwater to soak into the soil and reduces the danger of flooding. If you want to avoid lawn mowing you could plant alpines (some of which are good weed smotherers) or perhaps low growing shrubs (such a lavenders) or possibly some ornamental grasses which would only need to be cut back at the end of the growing season. Something like this would look more attractive than paving although another possibility might be to leave gaps in the paving and plant things such as Thyme and there are quite a variety of diffenent types which could be low maintenance.
Personally id say keep the lawn big and have a smaller paved area for seating and nice wide curvy borders. My garden is all borders and paving and with small kids its a no fun in the summer.
They need grass for slides, swings. Soft base under paddling pool etc
Id love grass but rented house so no choice!