Remove large hedge by own effort?

First of all, it should be clarified whether the neighbor is not a co-owner of the hedge. If so, you must obtain permission from him that you are allowed to remove his property. And cutting it down to a height of 20 cm, as suggested here, I would not do. Instead, leave as much of the trunk standing as possible so that you have a nice long lever arm.

... you can also shorten the hedge and additionally put up a fence - I wouldn't tear it out because that either results in incredibly hard work and expensive disposal, or you get someone to do the work and it becomes insanely expensive.

Why you want to have a fence on the planting stones up there, 'because of the children,' I don't quite understand. The children are much more at risk if they climb on the planting stones, but in my opinion, that should be handled educationally and not through coercive measures. How realistic your concerns about burglars are, I would clarify with the advisory office of the local police.

How about trimming the hedge and then using a tree root grinder?

In our district, the disposal of a trailer full of green waste costs 5 euros. It refers to a car trailer. We always bring 3m3. In the summer, I transported 9 vans including trailers. So that was almost 50 cubic meters. In total, it was 90 euros.

Above the hedge is our actual garden. The hedge marks the edge of the garden, where it then slopes steeply down towards the basement level. In the basement, there is a granny flat. My concern is that the children might crawl through the hedges while playing hide and seek with other children and then glide down the slope. That's why a fence, so that you don't suddenly stand in front of an abyss.

Fence: no idea what kind of fence it should be. I think here we need to make sure it doesn’t get too expensive. The primary concern is the protection of the children from falling down.