They use "cup" and "tournament" interchangeably. If you look at your schedule, the games will be marked with "C" for Cup. But if you look at the bottom of your league standings, there is something that takes you to the Tournament Standings.
If you go tournament standings.and then click on USA Tournament, Round 4 you will eventually find your game, but you click on "Next Games" several times to get to the page with your game. There are still 512 teams left in the USA tournament, so there will be 256 4th round games, and this requires several pages.
And just to keep it convoluted, if you look at your team's history, it will show your past performance in the Cup.
I don't think there is a fixed bracket. You probably keep playing D.IV and D.V teams since there are so many of them. 92% of the initial cup teams were in those divisions. And even though most of them were bots, they have won plenty of games over human opponents.