My doctor told me I had to be on the pill for at least a week before I would be protected, BUT it definitely varies depending on which pill you’re on and maybe other factors. You should ask your doctor. Or just wait a full month.
If you’ve both been tested, you’re using your birth control perfectly, and both of you are sexually monogamous (or are using barriers with any other partners you might have), then it’s fine to have unprotected sex, yes. Personally I double up on BC methods because I really really don’t want to get pregnant, but it isn’t necessary if you’re using your BC perfectly.
Hormonal BC has been known to help some folks with their acne, yeah. It varies from person to person, but that’s one of the more common side effects, from what I understand.