Edit armature with existing animations As far as I know, this is not possible, but I just want to be sure there's no way before discarding the idea completely. 

So, I have a character rig that's animated, and I want to use it with another model of a character with a different build and I need to adjust some bones in edit mode. After doing this I tried to reset the bones in pose mode with Alt + G, Alt + R, and Alt + S, but none worked. I tried to apply the pose, yet that didn't work either; the bones didn't change size. It looks as it should when I'm in Pose Mode and have it set it to Rest Pose, but then I can't animate it. 

Does anyone know if there's a way?
Editing armature with existing animations is totally possible.

When you edit the armature in edit-mode, you directly change the rest-position. Any transformation (animation) is relative to this rest-position, so by changing the rest-pose you might break the animation (but if the animation is mostly rotations and you change only proportions of bones there should be no problem) 

The rest-position is identical to pose-position without any transformations or constraints. This is by definition, so when you Alt + G, Alt + R, and Alt + S, and you turn off all constraints you must get the rest-position.

You can also apply any pose as a rest-position by Pose > Apply > Apply Pose As Rest Pose. How to do that with already binded mesh is here: Can I set my pose position like the new rest position?

Any animation can be also edited so it is not ruined:
How do I resize an armature without ruining its poses?