No, I don't cover it. I contacted Weber regarding a cover when I first bought it in 2009. They had a cover but suggested that I didn't cover it. They said it traps moisture. They said they only carried covers because they ad so many requests from customers.
I cover mine also and never have had a problem. Also the Weber covers are not very good, I bought an aftermarket cover off of Amazon and it has a vent on the back side for ventilation, works very well.
That could be. You'd need to let it cool down an hour or so after the cook before covering it.
About the only other thing you can do would be to somehow shelter the grill from rain. If the rain hits the grill, it's going to get into the catch pan. That's just how it is.