Going down the list:
1) Doable

2) Should be doable, I haven't done this yet specifically so I'm not sure if it can be done per ship (like, having one ship stop at 20 and another at 25), but I'm 99% certain you can tack on additional levels, ability ranks. Health, shield, armor progression is generally "increase per level", so it would rise naturally without needing any extra effort from you. Again, this is a little theoretical from me at this point. One thing you can't do is add more than 4 abilities - 4 is the maximum.
3) You can add more types, but only 5 can be assigned to the factory. We've tried increasing the number, or adding another capital factory with no luck. So for now, 5 different types in-game.
4) Doable

5) You can block phase jumping completely with an ability, but you can't have it based on planet allegiance, as far as I know.
6) You can add some damage effects, and ships already have it in their entity files, but I'm not sure on limitations of this.
7) Doable, already one big mod in progress that looks completely awesome.
8) You can add planet types. You can't add moons to existing planets and have them be functional. You can give a visual appearance of a moon by changing a planet's mesh, but it will be looks only. You can add a station structure, but it would also have some limitations (can't have squadrons and weapons, has to be one or the other, for example).
9) Not doable, as far as I know.
10) You can only add several more frigates to fill up the second UI row to 10, cruisers to 5, capital ships are already full. Cruiser/capital UIs don't overflow to the second row, haven't figured out a way to make them. Almost positive that's true for cruiser, definitely true for capitals.
11) Haven't looked at this specifically, but each diplomacy "quest" has tweaks you can do. I don't think you can do ship/planet gifting through modding at this point.
12) Yes, there's room left on the building UIs for several more buildings, and each building can have 3 different weapon types and most can have 4 abilities, so you can add some extras.
13) Building experience is not possible, I believe.
Hope that helps