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Here's some info on pet-friendly herbicides: https://www.gardeningknowhow.com/special/organic/homemade-pet-friendly-weed-killer.htm
and here are some more thoughts on getting rid of fox tail: https://www.gardeningknowhow.com/plant-problems/weeds/controlling-foxtail-weeds.htm

Basically there are two approaches, digging, and herbicide. It can be really hard to get every piece of root when you dig, which is why many people prefer an herbicide that kills the whole plant, leaves and roots. This article tells you more about the kinds of herbicide from which to choose: https://www.gardeningknowhow.com/plant-problems/weeds/conventional-weed-killers.htm
However, it also occurs to me, because you said sticky bushes, that you might be referring to the sticky quality of the bushes. If that's the case, your bushes, or possibly the trees above them, are infested with sucking insects that are dropping sticky honeydew. If you kill the insects, you can wash the sticky away with soap and water. This article will tell you more about these pests: https://www.gardeningknowhow.com/garden-how-to/info/what-is-honeydew.htm

If you are looking at a large area, probably you're going to want to use an herbicide. I expect a post-emergent will be necessary for the woody plants. You're also going to be looking at some continued treatment and digging for a couple of seasons. These articles have some more information: https://www.gardeningknowhow.com/plant-problems/weeds/using-herbicide-in-gardens.htm
https://www.gardeningknowhow.com/plant-problems/weeds/post-emergent-herbicides.htm
I don't know exactly what kind of home gardening help you have available in France, but if you contact the nearest botanical garden, they ought to be able to direct you to sources from whom you could learn of the products available in your area.

This article will help you:
https://www.gardeningknowhow.com/edible/vegetables/vgen/vegetable-weed-control.htm

You can use Round-up, but only in a very localized fashion. Cut the bramble back as best you can and then paint the fresh cuts with Round-up. The bramble will suck the Round-up into its roots and this will go a long way to killing the roots. You may have to repeat 2-3 times, but every time you do it, it will weaken the bramble.