I've got thousands of keys in my inventory, which makes it really hard to find a certain one that I need. If I remove all the keys that I already used to unlock doors from my inventory will I still be able to open those doors?
No, you won't. You need to be carrying them every time you want to open a door, because they get locked some time after you leave. It's just like when you pick a door once and when you return it becomes locked again. But don't worry. Keys have no weight and you don't have to choose a key to open a door. You just need to carry them. The game will choose the right key and unlock the door for you automatically when you go to open it.