First, let me say that the tournament is an excellent way to introduce new players to the mystery format because you get to play one season in each of the decades (60s, 70s, 80s, and 90s). And you will be in a league with veteran players and, usually, if you ask them, they will be happy to give you pointers on playing in the mystery format. On to your questions:
Are we setting up draft lists like you do for a 365 team, or are we each assigned a team and we get that team across the different eras?
Like other formats, you select the players that you want (within the 80 mil salary cap) and then the computer runs a draft and gives you the players that you drafted unless another GM selected the same player higher up in the draft. After the autodraft is run, you have a couple days to select players in a free agent draft, and after that, you can cut players and pick up free agents at any time.
Is each era a full 162-game season, 3 games a night, or some other make-up?
Each decade is a full 162 season played 3 games per night with playoffs for the 3 division winners and a wild card team with the best record. As usual, in each decade, if you win the league championship you get 2 free credits and one credit for runner-up.
I thought I was reading you may have a player on your team, but you're not sure which year he's from so you're not sure if it's a good or bad year until you're playing and trying to figure it out. Sounds interesting. That would support the latter from my first question that we get a frachise.
Figuring out the year for each player is usually critical to doing well in the mystery format. I say "usually" because one year a manager made it to the tournament finals without making any (or maybe just a couple) free agent moves; they got pounded in the finals, but still, that proved that you don't have to make a lot of moves to win.
Let's take an example. In the drop down menu for Info/Help you can access the Player Set Browser for the 60s. Now let's say that during the season Willie Mays is facing Sandy Koufax (a lefty) and he strikes out and is injured. Looking at his card you see that Mays is playing his '68 season. That is his worst season, so you may want to cut Mays and use the cash to buy some better players for your team. You can keep him because even on his worst season Mays is pretty decent, but if you do cut him, you can probably get more production for your money. After being cut, Mays would become a free agent and could be picked up by another team, but if that happens, Mays will still be playing his '68 card for the entire season. If Mays doesn't get injured, he might be hit-by-pitch which would mean that he is playing on his 60, 66, or 68 card. You can also try to figure out which year by looking at his batting statistics as the season progresses.