I will answer your questions in Order.
You earn your badges by battling other NPC's or Other Players in the RP. Underneath the Progress section, the first post, you should see your username someplace. If you click the spoiler button under there, it should show you your badge component progress.
In this league, you only have to earn 1 badge to particpate in the finals and you have to self assemble the badge by earning them through battles or contests via wagering. There is also a job booth setup to where you can earn them through story and performing difficult tasks. You need 8 badge components to create 1 whole badge.
While you can get into the finals with one badge, it's advised you assemble as many as possible, you can use additional SPARE badge's as do-overs or safety nets; in the example if you loose a battle and fall behind in the league, you can spend an assembled badge to keep yourself going in the league without having to start all over again and waiting an additional year.
As far as moves work; As I mentioned before, if it's possible via games Yellow, Red, Blue, Japan's Green Through X & Y (today), it's permitted. Move tutor, egg moves, tm's, hms and so on.