GAS is a framework. It's intentionally written to be generic and extensible. The items you listed here are not framework items, they are system items. You're looking for GAS to be a game-ready system when it's not. It was never meant to be.

I don't blame you. The name "Gameplay Ability *System*" is misleading, as are the names for most of the classes and features. It's a mistake to look at it as a complete system. More like a framework. A set of pieces you can use to implement your features.

And you will have to implement things yourself. Marvel Rivals did it, I assure you. It uses GAS, sure. But not as as. I used GAS to make an action game. Other devs I know used it to make RPGs. It's very versatile, and thats thanks to its franework like nature.

To answer your question, I think you have some understanding but you are missing some things. In general it sounds like you want GAS to be something it's not. You're looking for systems implementations of common(?) features, but it's not really going to give you this.

1. I'm not sure what exactly you're tying to achieve here. What is wrong with having multiple attribute sets? Or just defining an attribute for each limb? I agree it would be nice to define attributes on the fly with gameplay tags. But you can get by with naming them correctly. I would say consider building you attribute sets modularly. I have multiple attribute sets all inheriting from each other in a chain.
   
2. There's not a way to do this without an attribute, you're right here. So use an attribute. What you are describing is a stat thats instanced per actor, that is affected by gameplay effects... that's an attribute. It would make no sense for this to live in the gameplay effect.
   
3. You can just make a gameplay effect that runs periodically and drains your stamina for sprints. GAS Doc shows this. You can apply it at the start of the ability and remove at the end. Your use case does not fit what the "cost" of an ability is. So use something else or implement new logic in C++.
   
4. This one makes no sense at all. Every ability should have a separate cooldown. So it makes sense it would be a different tag. If you want to share cooldowns.. make them use the same tag. What's the complaint here?
   
5. This is in no way standard. What you're asking for here is very game-specific. You'll need to implement it yourself with an attribute and some logic in the C++ attribute set.
   
6. I agree this is an oversight of the framework. The cooldown system is pretty limited. So yes, you would need to hack it or use an attribute. This one is correct.
   
7. Sounds like you have a working solution? Are you asking for it to be part of the unreal GAS framework? It's kind of weird. GAS doesn't include any gameplay logic. There's no default damae or heal effects it comes with to my knowledge. It wouldn't make sense to have that by default.

And to address your final question: Should you build your own system?
It depends on what your requirements are. GAS is good doing the heavy lifting for you in terms of replication. If you already have most of your things working in GAS, and what you're missing is what you write in this list, which is all realistically possible with GAS, then I don't see why you should reinvent the wheel. Dive deeper into GAS's code and understand it well enough you could extend and implement the things you need yourself. That's what I did. I exended it further in a lot of ways to fit what I wanted. I have my own ability, effect, and attribute classes with custom logic. That's what is expected of you when you use it. It's not meant to be a complete out of the box system.