IF you dont have a safe orgun cabinetthen just keep it in a dry place. Dontput it in a case and justleave it somewhere. Sometimes the cases can hold moisture whichcan lead to problems.

I always take my guns apart and wipe them down after every season. Even a gun thats in my safe and doesnt get used. I actually enjoy cleaning\oiling the guns. I wouldnt treat the stainless barrel any different then a blued barrel in how ofter you clean or lube the gun. Its usually better to put the oil(or whateveryou use) on a cloth then apply it to the gun, instead of spraying it on and it not get it all wiped off.

There are many oils that work fine. Remoil is pretty cheap and easy to obtain(Walmart--Kmart) and seems to work good. Any lube thats designed for firearms will prob work just fine. Balistol is another good one and it even smells good.