The main reason to use quatenions is for the interpolation. Quaternions rotate in “straight lines”, regardless of what angle they’re rotating from/to. Like the path of an airplane trying to minimize fuel usage, their rotations describe great circles on a sphere. That’s not true of Euler interpolation.
You can make quaternion drivers, but it’s not straightforward. Mostly, using a driver is going to eliminate the main reason (smooth interpolation) to use a quaternion anyways, so you might as well not. Something with a driver is no longer interpolating (at least, for that driven value)-- it’s deriving the value, every single frame.
It wouldn’t be impossible to convert quaternion animation to Euler animation. I don’t think Blender has the ability to do so natively. But an addon could do it, so you might consider looking around for one.
Quaternions with Ws of n,-n are the same orientation, describing different directions of rotation. Rotation in the viewport should give you shortest rotations, from the sign of W, but you might end up with one that rotates a long path (>180 degrees) at the end, back to pose, when using NLA for example. If you interpolate to a 0,0,0,-1 instead of your 0,0,0,1 rest pose, that can fix some problems. But don’t try to just flip the sign in the sidebar, there’s not enough precision there. (Scaling -1 in graph editor should work to flip a sign.)
The smooth interpolation is why quats are the standard for bones (for objects, Eulers are the standard), which are often used for character animation. Some problems with use are due to Blender’s way of dealing with quats (normalized components). That’s being re-examined by devs, and we can expect some changes in future builds. One thing I hope and expect to see is shortest path interpolation for quaternions.