I think the ring you are talking about is the race for the diff bearing, moving it could affect the preload on the bearing. I took mine off to replace a leaking axle seal and got lucky, put it back on and no noise so far... that was about a year and a half ago. I didnt know what it was till I got it off.
As for the vibration issue, you mention that you just replaced the shafts (rebuilt ones I presume?) I also replaced both shafts on my 1995 2wd Legacy and had bad vibration under accelleration. I replaced them about 2 days apart and after replacing the first one didnt notice a vibration (but I wasnt looking for one) replaced the second one and it was very noticable. I assumed that the second one I put in was bad, so I took it back and exchanged it for a more reputable brand and there was a remarkable improvement, but the vibration was not gone entirely. I will be returning the first one I installed and replacing it with the better brand as well.