Re: Mold on the false back
Mold is a result of excess humidity in the hive. The feeder inside contributes to that humidity, of course.
The cure for too much humidity is improved air circulation. My suggestion is a top entrance:
If you don't want a top entrance, vent holes are another option. I built my TBH hives similar to Mr Bush's, with a top entrance, no other holes, and fed with boardman style feeders behind a divider without any mold issues.
USDA Zone 7A Elevation 1400 ft