To the 2nd point, I think we can settle that. The opponent will not know which criteria have been picked by the underdog and which are random.
To the first point, there are several problems with neither knowing:
1- I am involved, and at least I have to know all of the pairings, so it gives me an advantage (albeit I only have one character so a small one)
2- The ranking of the characters won't entirely give away the match-ups, but it will narrow it down. If we don't want the favored to know, I would have to do the ranking and not reveal them.
3- The underdog's advantage may be diminished too much if he doesn't know his opponent.