You can only see your profile when logged in.
Your crew, one leaderboard.
A league isn't a second game – it's the same Daily Quiz, just among you. You keep playing the same spots as everyone else; the league leaderboard just shows you next to the people you invited. Points, rank and streak all come from the quiz, there's no separate scoring.
You'll get an invite link to pass on. The only way into someone else's league is a link like that – there's no directory to browse.
A cup brings extra points:
Every league starts on the first Monday after it's created. Until then the table is empty – that's not a bug, that's the starting line.