For (1)- climbing, board games, team sports are all some ways I specifically do this for, but in a large city almost anything you find interesting probably has a group.
For (2)- I think it's worth explicitly going in with the mindset that you want to be social. For example, I go to the gym for some me time. I have zero gym friends (even though it is a good place to make friends, if you want to). So just showing up won't be enough IMO
For (3)- community needs investment. Most of the best groups I am in did not provide a lot of value to me for months. So you have to stick it out