Just recently the weather has become cold but.

Just recently the weather has become cold but not artic (lows 20 degrees) and I am having problems with my car during the warm up process. Ive had the battery die once and needed a jump during this problem as well so not sure if related. The problem is that the engine is running very rough, car shakes lightly and you can feel it kick at you (maybe misfiring). This happens for a good 10 minutes, sometimes less, all the while becoming less and less until it eventually dissapears altogether and I drive to wherever with no further issues. I am concerned and have no idea where to begin to fix the problem. Any help is very appreciated

2006 STS/V8 Northstar
Production: Saint Loraine Quebec