First, you're going to love it here. I think I can speak for BrewScandolo when I say that we will be happy to accommodate if you are looking for likeminded people to beer nerd out with when you arrive.
I see that BrewScandolo answered a lot of your questions, and I agree with him. I'll just add a little:
-what's hard to get here: I agree with BrewScandolo on this one. For example, I am dying to move to a kegging setup. After months of searching, the quote I got for a two-corny keg system with regulator, tubes, CO2 tank, and dispenser was 660 EUROS!
-where do you cook: For me, inside. I have a 13-gallon pot that straddles all 4 burners on my gas stove. Cooking gas here in Italy is fairly inexpensive. I also have a turkey fryer setup that runs on propane (US hookups). Propane tanks are available on post at the auto skills center (also US hookups). It costs about 35 USD to fill one (standard gas grill-sized tank, I don't know how many lbs.). The problem I have with cooking outside is how to chill; I live downtown, and there is no hose hookup outside of my apartment. I don't fancy carrying 6-10 gallons of boiling hot wort (or even cooled wort) up my narrow staircase and through my tiny door, so cooking outside just isn't an option for me.
-Where to store your equipment: Apartment size and location will depend on your OHA/LQA. I have plenty of space for my picnic cooler mash tun, brewpot, grains, other ingredients, and the hundred or so bottles I keep around.
-chest freezers: You can buy a Chinese model for around 200 EUR like BrewScandolo says. Keep in mind that anything you buy here will run on 220V and have a europlug.
What else to know: Yes, bring as much as you can with you, especially if weight allowance is not an issue. Ingredients are not expensive here, but equipment is!