Ok I am a pro at this, lol. My car sat in a pole barn for 37 years. Yes I had plenty of mice get in, snakes and other critters, lol.
I did put plastic down under the car. If you can jack the car up and put at least two 8" concrete blocks under each tire that gets the floor up away from moisture also.
Yes on the car cover. I always had a cover on my Mach 1.
Never leave a battery in the car remove it and wash the area with baking soda to kill any acid residue there.
I would completely drain the gas tank if possible.
You can mix up some oil and gas and squirt down carb while running to lube the valves and cylinders good before shutting off.
If you put jack stands under the car you can ruin your rubber bushings on the shocks and radius rods unless you block the upper control arms from dropping too low.
Now to keep the critters out. There are a couple places that they can easily get into the car. The most obvious is the cowl vents, cowl drains and a couple places in the trunk.
I have put pictures of the cowl and this is what I am doing during the clean up of my car and I have already done to my 73 vert.
I got 1/4" mesh wire from hardware. I zip tied pieces under the plastic cowl vent covers. I had the heater box out so I also put a piece over the cowl opening to the heater box. For the two drains on the cowl I push stainless steel scrubber pads into the opening. Water can drain but mice cannot get in. You do not have to pull the fenders just pull the plastic grills and reach inside and push into the drain holes. You might find a nest in there now.
In the trunk lots of times the rubber plugs in the drop offs get pulled out. Perfect place for a mouse to get in.
I use the electronic repellents in my 4,000 sq. ft. garage. I keep bags of corn in there to feed the deer and turkeys and have not had any mice in three years. You could put them in the barn and one in car also.
A mouse can just 3 feet up. One time we had a smart mouse that you could not trap in the kitchen. Me and my son sat and waited for it. It came out and climbed up the cabinets to get to the bread. We rushed in and it jumped off the counter and ran down the steps to the game room. On the way after it I grabbed a tennis racquet hanging on the wall. We cornered it and that thing was going crazy jumping up. So I gave it a good forehand with the racquet while in the air and bounced it off the wall, lol.
I think the mice use the dryer sheets to build nest with, lol.
You can see in the pics of the car the paint stayed in great shape. The aluminum trim did not. So go around the aluminum trim on windows and drip rails and tail lights and apply good coat of wax to help corrosion. I just finished polishing all the aluminum that was a chore.
The first series of pics is how the car looked after 37 years sitting in the barn and had not been washed. I have the interior out, suspension off and engine coming out when helper can show up. Got one more month to have back together to go to museum.