Strange symptoms, but the wayward/floating rear end is a typically cause by worn subframe bushings. This could also cause alignment issues, but is doubtful. Also, on the rear alignment, you are aware of course that there is no way to adjust or change the alignment on the rear wheels other than to install eccentric bushings in the trailing arm (part#33329058822). It is also possible that someone tried to install the eccentric bushings in the past and made a hash of it, it is also possible that the car was hit, or smoked a curb in the past causing the rear end to go way out of spec.
In any event, I would highly doubt that the above can be fixed on your garage floor and needs to be done in a shop. I had my rear end completely redone with new trailing arm and subframe bushings - it is a cow of a job and takes a good mechanic all day - budget $1k provided there is no damage to the subframe.