I keep my grill covered 7/24/365. There's always something to keep off -- sun, dust, dirt, pollen, bugs, leaves, rain, snow, ice.
My key is to use a cover that is loose and has good ventilation. I buy super cheap generic covers ($25 or less) that are slightly oversized for my grill.
Then I put a beer can chicken rack on the top of my lid before putting the cover on. So the top of the cover is tented -- the slope gets rain and snow to roll off and also creates some air space/air flow inside. Make sure the bottom is open and doesn't drag on the ground. If the bottom is too long, trim it with a scissors (do it in the back and the DIY won't be visible) or hook the cover up over the wheels to create some clearance.
You also want to make sure the cover had good vents -- for air flow and also to make sure the grill doesn't go airborne in a big wind. Like yours, my grill lives backed up against the side of the house. So I cut a hole in the back side of the cover below the cook box. Minimal rain/snow can get in back there, the hole lets in a lot of air, and the front still looks neat and tidy.
20 years, zero rust. Colorado is pretty arid in the summer (which definitely helps for rain, but not for sun) but also has lots of snow during the winter (which makes a cover mandatory).