I’m definitely putting mine away now, many don’t, but it entirely depends on your car and circumstances

I’m doing it as I’ve just had my MR detailed and I’ve only bought it for the summer.
In answer to the battery question, either will help but preference would always be to use a battery conditioner. The battery could well go flat anyway with it disconnected over the winter. I’ve wired the post connectors for my Ctek so I can connect to the battery in the boot without having to take any covers off.
Few points I try to follow for the winter:
use the brakes on a short run if you wash it before you put it away to dry everything off and save corrosion.
Also, leave the handbrake off, put the tyre pressures up to ~50psi, and roll the car to keep the tyres from getting flat spots.
Some people choose to do an oil change before the winter too, as as oil and gains surfactants it can cause contamination inside the engine... I’ve never gone in for this though.
I’m sure other people will add other bits too, hope this helps though
