Looking at running a team tournament

I'm not a tournament player in general. Is it a single day event? Five games in a day would be completely exhausting.

How does the army composition work between teammates? Are there restrictions on which races can ally? Does each player have a general (and BSB)? Or does each team have a single general (and BSB)?

You may try Triumph&Treachery book.

Also some different compositions for games (for an army and/or team).
For example: 2 games with single player each (2500pts), 2 games with "teamwork" (your variant, so 3k points per side => 1.5k points per player as rulebook allies) and final "mega battle" - 5k pts per side (with single (main) General/BSB).

There are possible variants&restrictions:
- limit on magic levels/mages per side;
- limit on magic items allowance (halve the pts on them/items just for General);
- "racial battles" - using similar Armybooks in teams (but different lists of course, with no doubles/mirrors) or even in tourney as a whole. Or "themed allies". Something like only O&G team/event ("WAAAGHomania"), only "Chaos" or whatever else. I remember we've used "Greenskin wars" - only O&G lists but all of them were themed (gobbos only/foot only/savage only/mounted only/etc) with loads of conversions&homebrew. The same with "Chaos".
- reserve/bonus pts to teams as a reward for game.
- % pts in list (25% or so) to change between the games)
- whatever else

About secondaries.
- Themed objectives as with WH40k. Give victory points for something like "destroy X units in [that] phase (magic/shooting)" or "successful charge [in X or over inches] X times per round" or "for each irresistible spell casted", etc.


So it'll come down to your group (available armies) and imagination.

I hope it helps.
Have fun.