That sounds typical for conservative Christian households, but is deeply unhealthy. It’s a huge violation of both trust and privacy, and also extremely, overtly bigoted. So, common? Yes. Normal? No. Healthy? Absolutely not.
(It’s probably not a majority of conservative Christian households that do this, but it’s not seen as bad in that community, and is therefore relatively common compared to the overall population. It’s also not unique to Christians. Many fundamentalist religions exhibit this sort of strict, spying-enforced behavioral control.)
Wicked people do wicked things. Good people do good things. But for a good person to do wicked things, that takes religion. (Paraphrased from Steven Weinberg.)
Then why are there more straight religious leaders convicted of child molestation than gay people? It sounds like your research lacked any actual data.