I lucked out with a conservative Christian dad whose love really was unconditional. Coincidentally or not, he was also not the kind of person to tease people in the way you describe. That sort of toxic-male ritual has always made me intensely uncomfortable, yet it's so prevalent in American society.
It's a sort of dominance joust, I think. I don't really know why people do it, but I've certainly never found it fun or pleasant in any way. Neither did my dad, evidently.
I think the issues are connected. Once he knew I was gay, he completely changed how he communicated about gay issues, even though he did not change his core religious beliefs.
I never had to doubt that he had my back.
I wish you and your dad had that kind of relationship!