If you want as many as 23 rounds then the link that Richard gives above is the perfect solution, since everyone partners everyone else exactly once and opposes each other exactly twice.
The link that I give above can form the basis of a 9 round schedule. Play the first 7 rounds as I suggested then rearrange the triads from the footnote to give rounds 8 and 9.
round 8 (A H W L) (B M T P) (C I U V) (D G K O) (E N S Q) (F J X R)
round 9 (L P V D) (O Q R A) (B C E F) (G H I J) (K M N U) (S T W X)
Doing this will meet the at-the-same-table-with-everyone-else-at-least-once requirement. If you are careful when you assign partnerships, it should be possible to ensure that players have 9 different partners.