Is there any impact on watch battery life with Garmin Connect Mobile constantly running in background?

Maximizing battery usually comes at some sacrifice, so if you have access to charger... yes you would likely want to keep the automatic sync with GCM/phone enabled.  I would think that if you had it off, but had wifi setup for at home/office/etc that it would still get some updates when near your wifi areas.  If I go on a super long ride or ultra run, often i will turn off wifi (optical HR, screen dimmed, turn off some sensors that I am not using on activity, etc) on watch.  Probably all marginal gains of time... but better than watch dieing 10hrs into a run/ride.