Hello! I'm glad you are enjoying Districts. Thanks for the feedback.
Before I answer the questions below specifically, I should mention that my goal with most of the design was to stick with the vanilla game behavior and UI wherever possible - everyone is used to that, and I want Districts to be minimally disruptive and consistent with the regular game (and fun, hopefully).
1. ... maybe it would be good to assign a color/border or something else, so that the buildings that can be built/viewed in Districts can be better distinguished from those that will be built in the city?
I have seen that those that need a District to be built appear blurred, but when the district already exists, I don't really notice immediately which will be built in the district and which will not... maybe I'm wrong...
Right, Districts are grayed out when a city *could* build them if a required district existed. I know what you mean, but am hesitant to add anything to this, mostly because the base game does not remind us or give other hints when new buildings or Wonders become available. Because the game doesn't do that, I would prefer Districts do the same.
2. ...maybe when viewing the city it would be good if the city borders also included the Districts, because they appear as if they were no longer part of the city's buildings/tile..
I think you mean that in the city screen, Districts are dark? (unworkable). Unfortunately this would be hard to do. The base game makes unworkable tiles appear that way, and Districts are by-design, unworkable.
3. The Districts can also be built outside the city limits, for example Districts for Wonders, but they are not active, which seems correct to me, but you can still build them!
Yes, this is by design. Just like you can build farms & mines outside of your territory.
4. ... Distribution Hub, it's great for what it does, but I'd like to be able to build it outside the city limits, how to say... like a distribution center but outside the city limits... without "biting" the city tiles!

It is posible?
- maybe a mini distribution hub inside the city limits that would distribute the surplus from that city to all the other cities...?
If you could build Distro Hubs outside of your territory, what if another civ built one next to it? Or, what if you built one in another civ's territory? What should happen?
It's unclear how the game *should* behave in such situations, and anyway it would be inconsistent with how the rest of the game works, unfortunately.
5. - I built a Distribution Hub one tile away from the city but... The Hub has 9 food units distributed, and in the tiles on the city border it has food: 5(potatoes)+ 9(3x3) from grassland + 1 from the hill - the 9 food units appear in all cities... its ok?
I really need pictures here, as this is somewhat difficult to follow. Please share those if you would like me to take a look.
6. Natural Wonders do not appear with resources:
I'm unclear on this one. Are you observing that Natural Wonders don't appear *on top* of resources? (e.g., Mt. Everest on top of oil). Or something else? Again, pictures would be helpful if you see an odd case.
7. I have a city stagnated at 4, without growth but the message to build a Neighboring District does not appear.
- I built a Neighboring District but... nothing
... the city started to grow when we entered Anarchy/Revolution to change to Monarchy...
Is that ok?
This is somewhat difficult to understand without more information or pictures. Do you have
maximum_pop_before_neighborhood_needed = 4 ?
Also, it's important to distinguish a city's (1) population increasing (e.g., from 4 -> 5) versus (2) stockpiling more food. So when you say started to grow I'm not sure which you mean.
Neighborhoods are designed to mimic the behavior of aqueducts and hospitals, i.e., you are able to continue stockpiling food up to a limit, but the city cannot grow in population without sufficient neighborhoods. If you see something inconsistent with that, please post pictures.
8. The message that tells you that you need to build the Neighbors District disappears very quickly, is not very visible, and is not repeated every turn.
Well, again the goal is to use the system the game already provides. The vanilla game uses that brief message system for "We love the King", day, disease, etc. so the speed should be the same. And I believe hospitals/aqueducts show only one "popup" message per city (ever) when growth is blocked by them, so in that sense, the Neighborhood-needed message is *more* obvious than hospitals/aqueducts, I would think.
As to the frequency of turns, when I first developed it, every turn felt somewhat annoying so I set it to every 4 turns. I could change that of course, but I also want to make sure people don't find it annoying.
9. The city asks me to build the Neighborhood District -> I build it -> the city starts to grow, then I destroy the Neighborhood District, the city continues to grow, how much it should have grown (one point).. the growth is reactivated if I rebuild the Neighborhood District plus another Neighborhood District (how many there should be).
So the population does not decrease and does not stop growing if I destroy the Neighborhood District!
I may reconsider this is the future, but I designed Neighborhoods not to reduce population if destroyed, just block further population increases. Maybe just imagine the people become refugees, etc.
About the "not stop growing" part, again, I think this depends on if you mean population increase or more stockpiling of food. If you could please share screenshots that would help.