I just watched the video again in full screen so I could get a better look at what is going on.
With gravity, I *think* the issue is just that the conversion from bone space to world space is messed up, so it thinks gravity is in the wrong direction. This should, hopefully, be relatively easy to solve. I'll try to take a look at it in the coming week or two and see if I can find a fix.
For the animated mesh, this issue is a touch more complicated. The target direction is relative to the root of the skeleton and not the bones themselves, so that's why its pointing in the same direction as the character animates. That said, even if the target rotated with the root bone of the jiggle chain, it wouldn't give the desired result of adding complimentary motion, or at least I do not think it would, as it wouldn't be taking the animation's motion into account when solving the jiggle joints.
I will need to do some testing on this and see if I can find a solution, but this may need to be a separate IK modifier since it will have to solve in different way. I'll see if I can find a 3D model with similar motion for debugging and will see what I can do. This fix may take a bit longer though, unfortunately.
Thanks again for the video! It is quite helpful for seeing what is going on.
Edit: After thinking about it for a bit, the animated mesh issue may just be a matter of rotating the target with the rotation of a bone, just not the origin of the jiggle chain (otherwise it would never rotate) but instead it's parent. If that is the case, then it may be easier to get the correct result. Will need to test to know for sure, but maybe the issue is easier to solve than I thought...