I just replaced my transmission mount, I think this is what you have referred to, it was town and causing some vibrations. I have not found any loose connection on that side. I unhooked my TCM took it out and stared at it (that's all I could do

)) ). It looked ok. Perhaps heat damaged it. I don't know.
About cold vs. warm. I don't know. Probably no difference. I think the sudden jerk happens when car worms up. I never had it first 10 minutes of drive.
I do not have CEL on. Once my CEL came on and it reported U0101 - Cannot communicate with TCM. After I unhooked the batter in order to get to my transmission mount the CEL cleared, obviusely, and hasn't come on. I wish I had a working TCM for an hour. To put it in and see if that fixed the problem. The part is too expensive to just throw it in.
What are your symptoms. Perhaps we could help each other getting to the roots of this, because this forum is not very active and we seem to have a problem at the same time.