I am not sure I would agree that a sense of belonging doesn't happen on mailing lists. I agree that they often serve more technical communities, but many people in those communities feel a sense of belonging. The nature of belonging is less about the medium and more about the engagement on that medium.
I agree. This is why for problem solving I typically recommend http://stackexchange.com/ as a better solution for more directed questions and solutions. It also makes generating reputation easier.
If you are looking for structured feedback from your users, I have found that surveys are really helpful. You can do this pretty cheaply with SurveyMonkey (http://www.surveymonkey.com) with free surveys up to 10 questions and paid for more questions. Every survey I have ever done has resulted in very insightful feedback.
My social media strategy is to register Twitter, Facebook, and G+ accounts. I generally post the same announcements to all channels but most discussion is on Twitter and G+.