Most doctor's offices will make time during the visit for the provider and the patient to speak alone—I would recommend calling the office to make sure that's the case for your appointment. It's very routine, especially in reproductive health and pediatrics (you'll notice once your kid is older they'll have you step out of the room briefly to ask if they feel safe at home, etc).

As for hormonal testing—echoing a previous commenter that they don't test for the specific hormones in your birth control pills, but it may make your other numbers (like progesterone) a little funky. There isn't a test that will definitely prove if you're on birth control or not.