I'm assuming your driveway is ashpalt. When I first got my classic and had to park my daily driver in the driveway, I would tend to park in the same spot. Part way through winter I noticed dents in the ashpalt from the tires. So you might want to watch out for that. If you do jack stands, make sure you have a piece if thick plywood or something under each stand, to help distribute the weight.
For my classic car, I have not used jack stand, instead I pump up the tire pressure to around 45psi, to help the tire keep it's shape.
If you disconnect the battery, you will lose more than the time on the radio. The engine computer will reset/lose its memory. So when you take it out, the computer will need to re-learn how you drive. When a car is new, the computer uses the first 500 (or 1000) miles to train itself. All that data gets wiped out if the battery is disconnected for a while.
While a car cover will keep the top from getting wet, that does not protect the underneath. But since you alrady drove it in the winter, then it probably doesn't matter. But if you care about that, look into one of those 'bubbles' (a big bag you park the car in).
As for car cover, I have a Noah for my classic. That is an outside cover, though it doesn't get used for that
But there is probably a step up that is more suited for outside. The key is to have something that is breathable.