Rod mat with plastic strips does not speak for a modern cottage garden. The pictures represent a colorful mix. Definitely find a uniform style or at least something matching.
Whether the wall is permissible afterwards I would not be sure. In my opinion, it is not listed in 11.2. To me personally, the fence with plastic strips looks more like a prison. To stay with the wall - who and how maintains the neighbor's side? - it would cost me too many square meters.
About the budget Fence once around the property leaves not much left in the budget. Therefore, think carefully about what you definitely need and develop everything step by step. You can do a lot with 20,000 euros, especially if you go towards cottage garden/natural style.
I would place playground equipment so that the children can always be seen from the house and the terrace. There is no peace at that age anyway.
Concerning the pool, provide connections and where and how you then run it, I would decide when the time comes. I would also only build it when the kids can swim. The budget must also be available again.
Plan the barbecue area in the hollow. If there is smoke smell in the house, so what. It will dissipate.
Really take one step at a time in planning. You can always still bring in your ideas and wishes or your gathered experiences.
I find fruit distributed throughout the garden quite nice. There are many ideas. Not only for fruit. For example, runner beans are sometimes used as a privacy screen or bean tipi for the kids.
When I look at the well picture, it fits into a cottage garden. But only individual shrubs, lawn edge, robot fits more a modern irrigation system.