There wouldn't be any problem leaving the battery on a 2 amp charge over night...The ideal would be to bring the battery in over night but cats may not have any tongues or fur in the morning if the're prone to liking batteries and chargers...Yes by all means get more fuel in the tank when you get it going.
...I know what you mean about cold...When I lived in northern Saskatchewan the truck of choice out there always was the Chevy 1/2 ton...I kid you not, I used to stock 25 to 30 extra door handles come November, December...it got so cold sometimes that the pot metal handle would break of in the customers hand....if you eve replaced a 88 and up Chevy door handle, they can be a beach...I got pretty good at it. And plastic bumpers, at minus forty, look at them sideways and they'd shatter. Also remember the a few guys that would bring their ice cold cars into my shop and from -40 to +75 in 0 seconds flat, replaced a few side glass, door glass and even windshields if they had a minor chip or hair line crack...they would shatter...did I mention I hate cold weather. LOL
Good luck my friend