I would model teams as vertices of a graph and quizzes as directed edges. Each edge would be decorated with the round number. Then the problem becomes one of creating a graph with uniform in and out degree at each vertex, ie, a kind of directed regular graph. Furthermore the round numbers must be picked to satisfy all the simultaneity contstraints and the constraints imposed by mapping related vertices to rooms. It is a mess :)
in reply to Round-Robin with Three-way Matches
In complicated problems like this, I would assign each violation of a constraint a numerical weight. Then every configuration of teams, quizzes, round numbers, and rooms would be graded by the sum of all the weights of the violations. Then the problem becomes one of minimizing the total weight over all configurations.
To minimize, I would use a genetic algorithm (e.g., Algorithm::Evolutionary) or simulated annealing (Algorithm::Evolutionary has some simulated annealing capabilities as well). In generating mutations, i.e., new configurations, it is best to satisify the truly hard constraints automatically. So always create mutations that have the same number of one-quizzes, two-quizzes and three-quizzes for each team.
Can't help you with the kitchen until you add more geometry to the spec :)