maestroruffy: GOG connect is actived only in specific periods (Christmas sale, summer sale, maybe because thay have release a new game, etc.). I think that your profile should be public to allow GOG connect to know if you have a game in your steam account. When GOG Connect is not active, I think that it is not needed to have your profile switched to public (why should it be?). The last question I think you wonder if it is safe to have your profile public. I think this questions is more related to privacy than security. You should not have security problems if Steam works as it should work, but anybody would have access to the information that is in your profile (games that you have, badges, etc.).
In the GOG Connect page you have a faq, but if you have more questions it is better to ask support, because some of the questions are a bit technical and I am guessing the answer :-)