I would make sure you clean around where the flapper sits. Make sure you have the correct parts for that toilet. Sometimes the universal crap can cause problems. Go to a plumbing supply for parts.

1. Eliminate standing water sources in your yard and around the outside of the house. Even the smallest amount like a coffee cup lid can breed larvae.

2. Install a quality bug zapper outside house.

3. They are not normally used in residential homes but they do make powerful fans that blow air out of the structure when doors are opened.

4. Rub yourself down with garlic.(just kidding)

Nope. The plant thing can happen. Pull em out, spray root kill if you want. Clean away loose gunk, don't let it go in the pipe. Install new stuff.

I think cement board is always best for moisture resistance

You could attach 3/4" plywood horizontally to the studs, then mount the bracket to the plywood. Further, if at least one of the studs works, plan the plywood mount where both the plywood mounting screws, and the bracket screws all go in one stud.

I had a rental property that prople smoked a shitload of weed in. Got the blu-dri hepa scrubber from home depot rental. Rented it for one week..took the whole week. 300 bucks thereabouts. It worked however