Please see our post: I is not an object …
You have provided several examples to ask your one question. This makes the answer more complex than the one word responses you received on the Web site.
What you ask is, effectively, which should be used as a subject, I or me? “Betty and I (subject) are going out.” is correct here. “Betty and me are going out.” is incorrect. It is not surprising to us that you have heard incorrect usage on TV. We could likely make a living correcting grammatical misuse on TV.
Now, when you move on to your … “Or join Betty and me.” you have changed the question … this is correct because, as we mentioned in our earlier post, “I is not an object …” In this case, me is correctly used as an object.
BTW, we recommend spell checking the title of your post: “… English/grammer”
Thank you for your (unknowing) contribution to our blog.