Hi and welcome! :) Yes, the weekly socials are open to anyone. If you have specific questions, I recommend contacting the group organizers directly, as they are more likely to respond via email or Facebook.
Sure thing - I just assigned you as an organizer to the group. You should have access to edit the group's info and events.
We have a contractor who cross-posts EA events from Facebook and Meetup. If you'd prefer to manage your group's events yourself, we can ask the contractor to skip your group.
LessWrong doesn't have this event cross-posting. Possibly it's less relevant for their site than the EA Forum.