I have registered an account on SDF's Lemmy server. Can I use this account to log in to other servers, or do I need a separate account for each server?
No, you have to make them per instance, but its sorta redundant as each instance communicates with each other and displays the same content / can communicate with each other (example: You're lemmy.sdf, The user below me is Beehaw, and I'm shi.tjust.works
That's mostly true, but there are a few exceptions where certain instances choose not to federate with certain other instances. It is also possible to make an instance that does not federate at all and therefore would be locked to that single instance.
I'm pretty new too, but as far as I can tell you can't do that. I just tried to search for a lemmygrad community through beehaw and nothing showed up, which is about what I expected with my understanding of how the fediverse works!