Re: Alarm going off and doors auto lock issue
I would like to help, but my GD3 was wrecked at 18K miles (without your issues) when a meth-head in a lifted pick up pushed my it off a cliff with me int it. Fit was totaled and I healed. So far, no such issues as yours with my GE, other than having an air bag that shoots people to death.
I suggest that you visit an auto parts store and ask them to attach their meter to the OBDII port in the dash in from of your right knee and see what, if any, codes show up. Or, visit a local mechanic (NOT a dealer) who specializes in import car electrical repairs. He will have a code reader, and the ability to find the cause of your weird symptoms.
Replacing the battery, if done right, should have had absolutely nothing to do with your symptoms.