Hello, gt50s --
I agree with Shelly (as I usually do!) on the issue of medication. My case is different from yours, I realise, but it also struck the right side first and gave me a tremor. When I got cold or nervous, the tremor shook my whole arm and my insides, as Shelly described. And, as you said, morning is the worst time. Rasagiline did not have any effect on my tremor. For me, it was Mirapex that nearly eliminated it. I still have a slight resting tremor sometimes, but nothing that is really noticeable to others.
On the question of PD's attack on the second side of the body, I suppose that is also an individual thing. But I can at least say that in my case, it is very slow. I thought I was detecting left-side symptoms eight years ago, and today I'm just beginning to be certain of it. Sometimes my right hand out-performs the left hand, as when I wash my hands or play parallel notes or runs on the piano. Therefore, I'm pretty sure both sides are affected.
I'm still taking Rasagiline, too, by the way, because my doctors believe it slows PD's progress. I am doing well, so I won't rock the boat at this point.