As for gender roles: Fidelio is unquestionably the Alpha Male in our pack, while I am unquestionably the Alpha Female.
As for who does what, we make those decisions based mostly on skill set. I'm the shopper. He's the plumber. I'm the computer guru. He's the mechanic. I'm the gardener. He's the winemaker.
We both clean house, because neither of us really like to. I'm the primary cook, but he makes the best meatloaf.
When an ordinary decision needs to be made, we go with whichever one of us has the stronger preference. For major decisions, we discuss it and come to consensus. When we can't reach consensus, we table the discussion and come back to it. On those very rare occassions we just can't reach a consensus, we usually go with what Fidelio wants, because he is: 1: super smart, 2: widely experienced, and 3: usually proven to be right in the long run.