There is no simple formula for making the schedules that you want, instead you would need to create an algorithm that optimizes the schedule to your requirements (by searching a lot of different ones). If you can play enough games, then for some numbers of players there are balanced solutions for 4-player games. The 9 players and 2 games is one such example, as you can take
the Whist schedule for 9 players, and ignoring the 2 vs 2 aspect, consider the schedule to be simple 4 player games, then every player would play with all others exactly 3 times over the 9 rounds. Other Whist schedules that you can find on the Wiseman site and here by following the "inspired the forum" link at the top of this page may also be useful.
Two questions. Are you wanting to play games simultaneously in rounds (so in each round two 4-player games happen at the same time)? Are you willing to play more games to achieve a balanced schedule, or are you really looking for the minimum number of games so that all pairs play together at least once?