I see no reason why you can't still get a great training effect while "injured."
1. Hit your lower body HARD. As you alluded to, deads, FRONT squats, lots of unilateral work.
2. You can't really go wrong with more scapular stabilization/retraction work (ie: various rows). 6x6, 6x8, etc. Probably wouldn't be a bad idea to add in some extra ER work (prone trap raise, band ER, etc).......but this will be dependent on what your PT actually says what the issue is.
3. And yes, you can still train one side/limb and still have it neurally "carry-over" to the other side.
4. It's been shown that loss of LBM doesn't start to kickin till 3 weeks of detraining have occured. So you won't lose as much as you think as long as you're still training. Basically, you're not going to shrivel down till you resemble a 13 year girl cause you can't do any pressing for a few weeks....
Again, you I/we can't really make any suggestions without really knowing what the issue is. But if it is what you THINK it is, then I see no reason why you can't do the above. If it's just tendonitis, it's just a matter of eliminating abberant movements and loading up on some NSAIDS for a few days and ice.