Stabilizer goes before storage. You can always drop a bottle of seafoam in the tank to help with the moisture that might have occurred, but I wouldn't worry about the stabilizer at that point.

 

Completely removing a Battery from late-model cars with Security Systems designed to thawart Vehicle and Accessories theft can cause a myriad of headaches for owners and repair shop techs.

Depending on the vehicle:

Starter and ignition systems may be disabled.
Accessories rendered inoperative until secret codes entered to restore functioning.
Some cars with anti-theft systems may become disabled requiring the vehicle to be towed to a dealership for re-booting or reprogramming to function again.

A very simple solution to avoid having trouble when replacing the Battery is to provide voltage to the system during the swap.

You don't need much.

Connecting a fresh 9V Battery to maintain +B Voltage to the car is a widely recommended fail-safe.

Hook-up the 9V Battery to back-you-up BEFORE disconnecting the car battery!

Get one of these.
Click Amazon link.