How should competition data be architected for spreadsheets?

Question

I could use the help of some spreadsheet experts. I am attempting to use spreadsheets to track competition results and am running into stumbling blocks.

Input Data:

1. I have a list of tournaments. For any given tournament, a number of teams may show up, and each team may have some group of players.
2. Teams change constantly: Michael Jordan may be in the Titans for the January tournament and the Spartans for the February tournament.
3. Teams don't play against each other directly, players do. A team's score at a particular tournament is the number of matches their players won.

Requirements:

• For a particular tournament, show all team's scores
• Calculate all players' career stats such as matches won, points scored, matches won in particular tournament types, and if possible win rates against other specific players
• Easy (non-duplicate) entry of new match results

Current Setup: I have attempted to tackle this problem myself by organizing the data as follows.

• "Tournaments" worksheet where the columns are
  • Tournament Name
  • Tournament Start Date
  • Tournament Format
• "Matches" worksheet where the columns are
  • Tournament Name
  • Round #
  • Player 1
  • Player 2
  • P1 Points
  • P2 Points
  • Winner (calculated by comparing P1 Points to P2 Points)
• "Teams" worksheet where the columns are
  • Tournament Name
  • Player Name
  • Team Name

This allows me to successfully have a "Player Records" worksheet where I calculate player's lifetime match win rates, point percentages, and such.

Issues with this design:

• Pivot tables looking at player data do not function correctly because Player 1 and Player 2 data are in different columns
  • Whether a particular player shows as Player 1 or Player 2 in a particular match is arbitrary, but any use of pivot tables seems to limit me to using only one of the two columns as "source"
• Not a good way to calculate team scores for a particular tournament
  • I am unsure of how to write a formula which looks through all of the Matches worksheet, finds only records for the relevant tournament, then links the name of the winner to the team name via the Teams worksheet.
  • I tried to add a "Winning Team" column to the Matches worksheet, but I need a VLOOKUP that checks against two arguments, not just one: it needs to find the "Michael Jordan" entry that corresponds specifically to the "January Tournament" entry in the Teams worksheet.

So, the big question is, is there a better way I can organize/input my data so that I can use pivot tables to gather player results and automatically calculate team scores in tournaments?

I think that's all the info I can provide. If you made it this far, thanks so much for reading through this. I think it is an interesting problem and hope someone else can give some guidance. Much appreciated!