Full random JC

Am I correct in understanding that the game continues until the first defeat, and there will be no lower bracket?

Yes, except for the final.

Will all players participate in the first round?

It all depends on the number of participants. If it's 2^n, then everyone will play. If not, someone will get a free pass, but in the second round, there will be 2^(n-1) players, and everyone will play.