You can bake the jiggle deformer to point-level animation (PLA) by selecting the object that’s being jiggled in the Timeline and running the Functions>Bake command with PLA enabled. Then the offset should work.
I can’t really tell what’s wrong without seeing the scene file, unfortunately.
You might want to check out the free Magic Car plugin by Nitroman. I believe it’s just a sort of autorigger using Cinema’s dynamics system. You can get it here, or check out the video tutorial: https://nitro4d.com/product/magic-car/
That’s a tough problem because cloth dynamics and rigid body dynamics don’t really work together. You may be able to Mickey Mouse it to look right by using dynamics connectors to control the rigid body objects while belting them to the cloth. Here’s a basic example: 10729_sticktocloth.c4d.zip
You need to be in animation mode when you move the camera (Tools>Modes>Animation). When you’re in animation mode you can move objects and their keyframes will come along for the ride.
Another option would be to put the camera in a null and move the null instead of the camera.