Maby your body is cold and somwhat shivery so your not shifting like you would if you were comortable.
But seriously, yes the mounts could be bad and the oil thick when its cold. You might try a different oil thats less seseptable to the cold. Also some of your simptoms like improving when you let off the brakes could just be that your engine speed in relation to the trans speed happens to mesh better at that particular time.
If i drive my porsche like i would a normal car i would get nowhere. I almost always blip the throttle between shifts and heel toe when im downshifting and it works just fine