Yes, some people do complain about that. The short of it is that Panasonic GX1 lacks in body stabilization that all Olympus MFT bodies have. You can eliminate that in part by keeping the shutter speed faster than 1/250th of a second, or buying lenses that have Image Stabilization (Lumix 45mm-200mm for example).
I hope that answers your question.
I'm not sure I understand what you mean by "digital zoom"?. With the 14 - 42mm X lens with electronic zoom I can use "manual focus". I have not tried other auto lenses which have the manual zoom ring in addition to the electronic zoom.
Prior to my purchase I read a lot of negative comments about the battery life. Personally, up to this point, I have not been disappointed. I did purchase 2 spare batteries, but so far, I haven't had to use them. I don't shoot for hours at a time, but still, off and on, it's not bad.