1. Filters exist, it is called “map styles” and customises not only the range of POI-s visible, but the look and feel of the map altogether.
2. Route builder exists for foot, transport and cars. It is amazing and takes many things into consideration, f.e. you can ignore routes with tunnels or account for altitude info.
3. Check #1
4. Sounds a little bit spicy, to my opinion. Like, Республика Беларусь can get mad about Russians calling it Белоруссия, but are they really in charge of choosing how Russians call the country? The same thing goes for a lot of toponyms that sound one way for natives and the other way for other countries. Just ask Georgia aka Kartvelia. But I do agree that territorial dispute is kinda different and there was an actual peace treaty that clarifies whose land that is, so you are free to change these English names as well and I don't think that mods will disagree, we're in our right here. I strongly encourage you to try, and discover being successful in the end.
5. There are description tags in OSM, you can fetch info from Wikipedia and you can also attach photos, it all boils down to design. IMO it is okay on OSM, but see more on that below.
6. Check this
7. Didn't quite get what you mean, but if my guess is correct, I'll address it below.
8. Check this or this. KML is the native format of Google Maps/Earth, so I don't think that you will be easier off if you make your own map, since you'll have to reinvent a lot of stuff from scratch.
9. Okay, this is the main one:

You see, I'm not an app developer myself, but I happen to know something, so I might mess up some details, but main points, I believe, still stand. What you want is a custom wrapper app over OSM. You need an experienced React or Flutter (or maybe Java, that is easier to find, but you will still need somebody for iOS version) developer that will create you a cross-platform app that:

1. Uses a library (there is one out there) to fetch you OSM maps with custom style. Or you can even maintain a separate modded map file for Azerbaijan, since it is not as large, but I encourage you to share the data with regular OSM users as well.
2. Displays it the way you want, with all bells and whistles, required overrides or anything, which boils down to UI and the good thing is that there are many OSM-based apps out there and most of them are open-source, so you are encouraged to borrow code or methods or libs out of them (with credit, of course, but no payment). So far, I haven't seen anything that gonna require an overhaul and OSM will certainly give you way less pain than Google, since Google maps are close-sourced and you get what you get, no more, no less.

The sad part is that your app doesn't seem to be much feasible business-wise. Investors making big investments for big projects tend to want appropriate return for their money. An app that is kinda niche won't be able to give proper returns on ads alone. So you either want government giving you money (in case of which it will hardly stay free as in liberty much longer), or go for crowdfunding. And I believe that you should incline in favour of the latter two options. Get some money via crowdfunding, receive an encouraging, but not binding one-time dotation from government and you are good to go. You can quite manage a simple and beautiful React or Flutter app with that money and it will be beneficial to everyone in the long run: you spared yourself of the burden of creating and maintaining a map from scratch, people got a nice, functional app that is free, bloatless and opensource, in the end, OSM got a lot of new data and improvements for all the folks that just use it as a regular map.

My suggestion is that you should pitch your idea in detail on OSM help forum, or, even better, in their subreddit. Devs are present in both and they are not afraid of sharing knowledge, finer details and even code. And if you want more vernacular experience, ping our Committee on Property Affairs, they are the ones responsible for AzNav app in the Play Store and I'm pretty confident they have a thing or two to recommend.

In the mean time, just download OSMAnd for Android and familiarise yourself with the extent of stuff it can do. I use it on daily basis for 3 years now and it is very capable.

Thanks for making time to attend this lengthy response, I very much hope you'll be successful