I am sure if you wade through all these posts you will get an answer for the first part of your question. In reply to the last, Harley has redesigned their dressers for 2009, with a new frame and engine mounting system. The few reports I have read suggest they handle better than an unmodified previous model.
Back to that wobble! The rubber-mount engine, transmission, swingarm and rear wheel are mounted as an assembly into the frame via three rubber mountings, one at the front of the engine, the other two behind the gearbox, one either side. These two have a spindle going through them, which also goes through the frame, back of the gearbox and swingarm bushes.
Being rubber, there is a small amount of sideways movement possible in these bushes. In addition, on earlier models like my 1990 FLHS, the swingarm used flexible bushes where that spindle passed through. So that makes two sources of sideways movement. Later dressers, I understand, have solid bushes in the swingarm.
The front engine rubber mounting has plenty of sideways movement in it, but Harley use a stablizer link to stop that. There is another stabilizer between the cylinder heads (on the front head for 2009 models). Buell models all use a third stabilizer link at the rear of the gearbox, as do the current rubber-mount Sportsters, indeed Erik Buell patented the system in the 1980s. No idea why Harley hasn't used it on all their rubber mounts!
On my bike I have replaced the flexible bushes with solid ones and fitted an additional stabilizer, which goes under the rear of the gearbox. There are probably a dozen brands now, selling these things, but I used TrueTrack. Most of this thread is about the Bagger Brace, one of the alternative choices, all setting out to do the same job, in their slightly different ways. I also fitted a TrueTrack kit to our Dyna, which does not have either front or rear stabilizers as standard.
The benefit is greatly improved handling, both bikes being far more stable in all circumstances, even just riding in a straight line, but especially around bends. They change line better and are a real joy to ride. If you cannot find someone with a modified bike to try, just get a kit and fit it - soon. You will not be disappointed!
Hope that helps.