Since art is something personal to everyone, some will classify games as art. That said, personally I've never played a video game I've considered art (visual novels, on the other hand...)
VNs are considered games by some people. Im still not 100% sure what to classify them as myself.
But VNs hit much harder, maybe its just because they can handle topics that wouldnt be suitable in games, and unlike an anime you have input, so it gives it more impact if the [censored] hits the fan becuase of your choices.
Im guessing your thinking of Clannad ?
Speaking of great VN's try G senjou no maou.
Gameplay isn't art. The music is art itself, it doesn't make Skyrim a work of art, maybe a collection of artistic pieces - like a museum.
The world design could be a form of art I guess, it depends what genre of art you try to fit it into.
Eh im not so sure. Not talking about skyrim in particular, but I dont think you can remove parts, leave them alone and have the same effect. The music and visuals are all part of the product. The impact together can be much greater, and you need the game to bring them together.
Gameplay for the most part sure it may seem meaningless, but if the atmosphere changes in a certain way, and its becuase of your actions. For me, it makes it hit harder.
If you just got a happy ending, sure thats great, but its much better if you had to work for that.