TL;DR: you don’t have to worry about it half as hard as you are. It will just work.
Still, it’s good to get some understanding of this stuff, so:
A better way to put it is that quaternions rotate in single axis rotations, while Eulers have “torque”-- if you have an airplane rotating about the center of the world, the quaternion interpolation will give you a great circle that they actually fly, the shortest path between two points, while the Euler will curve somewhat. How much, depending on the exact points chosen, and the airplane’s default rotation and axes.
Not sure what you’re asking-- to change a bone’s transform to Euler, you set a bone to use XYZ Euler, in properties/bone/transform. Is that what you’re talking about with changing rotation method? In that case, what is it that’s not being set to it?
Edit: oh wait, maybe I get you-- are you looking at the rotation mode in object mode vs pose mode? In pose mode, it’s showing you the rotation mode of the bone. In object mode, it’s showing you the rotation mode of the entire armature. These are two different, unrelated things. If you’re not rotating your armature, your armature’s rotation mode doesn’t matter. (And personally, I prefer not to rotate armatures, I like transforming a root bone instead.)
I’m not sure, and it may depend on the export function, but it really doesn’t matter. Quaternion and Euler angles both represent the exact same orientations at their keyframes-- you can easily convert from one to another. Where they differ is in the interpolation between their keyframes. I don’t think there are any formats which specify interpolation, that’s entirely up to the engine that’s using them. So if you use a .blend in Unity, say, then whether your animations act like quats (normalized component like Blender uses, or slerp quats) or Eulers (any of the six flavors) depends entirely on how you tell Unity to interpolate-- at that point, it has nothing to do with the original .blend armature.
If you only have one keyframe for every thirty frames, you may see differences. But what most people are going to do is to sample their animation at 30hz-- ie, they’re going to bake their animation. And remember, it’s only interpolation that differs, not keyframes. Are there going to be differences between frames when your game is running at 60hz? There will be, but they’re likely to be minor, because the rotational changes at 30hz are small, and because it’s unlikely you even really looked at your Blender animation that carefully anyways-- it doesn’t even show you 60hz animation at default settings.
It’s not yes, but I’ll answer the question anyways because it’s useful info, and not just for this case. Select all your bones, with one bone active. Change the rotation type. Right click in the field and select “copy all to selected”. Not so hard.
There’s probably an addon, but you can do that without an addon by duplicating the bone or object, changing its rotation mode, giving it a copy transforms constraint targeting the original, and then baking the action. I wish Blender was smart enough to just update your animation with rotation mode changes-- yes, you’ll lose keyframe handles, but that’s invalid information with a rotation mode change anyways.