A whist design gives you most of what you want, everyone as a partner once, facing everyone exactly twice. To see an example, click on "Schedules" above, then select whist and 16 players. Each foursome is an adjacent pairs of matches, that is foursome1=match1 vs match2, foursome2=match3 vs match4 etc. The only problem is that player 16 is always scheduled to start in the first foursome, so you could try rearranging the order of the foursomes within each round to get more balanced starting times.
