I don't think a "draw" feature is present in the game. You'd have to run a draw using a random number generator or something similar outside the game and apply it yourself - you can use the "Edit Playoff Matchups" for each round to adjust matchups and simulate a draw.
Set # of regular season games to 0 so the game doesn't try to set up any kind of regular season / group stage. You can also include byes, but you have to include teams that won't actually play for it to work correctly (the game won't let you just schedule a bye for a team initially).
Here's a really simple example of what you can do with tournaments. As you can see, San Antonio and Philadelphia received byes in the first round. Then Brooklyn drew Dallas and San Antonio and Philadelphia drew each other in the second round. I had to have 8 teams in the tournament (while only including 6 in the actual bracket) for the byes to work correctly - I would test this outside the main save you plan to use it in before implementing as if you get the math wrong on # of byes in each round the game will not run the tournament properly.
As for scheduling it concurrently during the season, I haven't done the legwork on that but I assume all you would need to do is tinker with your schedule files so that no teams are scheduled to play league games while the tournament is going on.