FC stands for Free Company. Those are basically FFXIV's guilds. Also, if you're ever getting into hunts, try finding hunting linkshells. That can be very helpful, but depending on the server, there are times when no one is active at all.

I'm from Jenova myself and although I see A LOT of people in the three starting cities (more than I'm seeing in Ishgard now that I finally got into HW), including new characters, I definitely don't see that many people doing FATEs. That could be a thing with earlier content, as I remember seeing lots of low-level FATE runners when I tried the 14 days trial about two years ago. Anyway, I'm starting to level AST now and I'm doing that by running dailies and levequests with my chocobo. The chocobo has also been a great help with FATEs in Eastern La Noscea. The ideal approach would be trying to find people who you can level with, so that you can run dungeons and FATEs more efficiently. Leveling alone sure is slow.