Depends. I don’t know much about the ActivityPub standard yet, but on my matrix instance I could set it up without federation, or with federation and a blacklist (blocked nodes), or with federation and a whitelist (allowed nodes).

So it depends on your setup, and the setup of other instances.

There’s no need for them to approve anything. You just run your instance, subscribe to the various communities and, perhaps, create some of your own. Any posts to those communities will automatically go through.

Short answer: You will need to federate your instance with at least a few others, but it goes two ways, and yes, other instances federating with yours will need to meet the approval of that instance’s admins.

When instance A is federated with instance B it means A will copy all posts and comments from B, in addition to those created locally.

When you create your own instance you can federate from any other instance and you can start receiving copies of all posts and comments they have. [This is not done in bulk; rather, it checks to see what communities your instance’s users are subscribed to, and only copies posts and comments for those communities.]

However, in order for posts and comments created on your instance to go to others, the admins of those instances need to allow it. As we’ve seen from the recent kerfuffle between beehaw.org and lemmy.world, this may not always be automatically allowed. As more and more refugees are coming in, admins of large instances can be overwhelmed and may decide to be careful of small new instances until they’ve proven they’re well-run.

Yes, once you have a server and domain you can read the lemmy docs and set up an instance of your on that server at that domain. You can then make an account on your own instance, and from that account do all your subscribing and posting and commenting on all other instances which haven’t defederated from your instance, which at the start will be none, and will probably remain so unless you start doing something unsavory. You can open your instance up for others to register or not. Regardless all communities created on your instance will also be available to every other instance (again, barring defederation).

selfhosted should have more helpful answers

Most of the servers running lemmy have federation enabled and open (as you can see, I’m one of those self-hosted instances) and you can just interact with them. Some do have federation locked down, but that’s the minority