2004 GMC Envoy P0128 and Losing Coolant

I bought the envoy a couple months ago with the only known problem being the ac resistor (a/c only ran on 1 and 5) fixed that
Then the battery kept dying
With bad battery I noticed it seemed to surge slightly when going and when idle rpms drop from 600 to 400 when ac or defrost is on.
Replaced the battery and now it will die if you idle with ac on.
SES light was on when I bought it so I got the code P0128
Engine temp never rises above 160-180 on the gauge
Also noticed I keep filling the overflow reservoir with Dex Cool and it disappears (not sure how long it takes to empty) The radiator is full.
After looking the coolant is constantly dripping from the overflow tube into the passenger fender well.
Plans are to change the thermostat (and temp sensor while I'm there) and clean the throttle body.
My question is could the problems be connected and Why would the coolant leak from the overflow with reservoir empty with car cold or hot running or not.