Shouldn't be a big deal to leave the battery connected if you're going to move the car every now and again. You should move it or you can get flat spots on the tires.
Also, if it's cold stored I'm told you should store it with either an empty tank or a full tank. A full tank won't have much room for condensation and when you fill the empty tank up any condensation that did occur would be masked by all the fresh gas.
2.5 months isn't an extremely long time to store a car IMO. The car may not recognize your smart key when you go to get in it if it's locked when you grab the door handle. The cars go into a "sleep" mode when dormant. Don't worry though, once you click the unlock button on the remote and the car is started all will return to normal. You might want to consider a trickle charger to put on it while you are away which you can find at most parts stores or we have them for $65.00.
I typically only go 2-3 weeks without driving mine due to winter weather here. I purchased a nice trickle charger from Griots Garage (unfortunatley no longer in stock) that keeps the battery up. Works great.
The tire flat spot may be your biggest issue (as I learned from other forum members).
I agree about the gas tank, I would keep it full, but one could make the argument to keep it empty for that long of a time frame.
Yea I got the brand new porshe 911s with PDK and I got it 3 mouths ago. I bought a trickle charger and put small cut out peaces of rug under my tiers and there PS2s and also ,put up the teir pressure to like 55 60 psi and never get any flat spots. Try that
Keep the tank full, use fule stabilizer (Stabil is good). As far as the tires??? Not sure what the best is for them, I thought about jacking the car and putting it on stands. The idea about over-inflating the tires sounds like it should work, but I would be worried about undue stress on the tires. I'll check with a tire guy that I know and post what he says to do.