Sorry there are too many variants here to consider them all. It is actually impossible to meet your criteria if the number of players is 21 or more. For example with 21 players there are 7 who will have only 6 games (the ones with a bye), and those 7 players can only play on court with a maximum of 18 others, so there must be at least two players who they never meet. For me this makes 20 players the most interesting scenario, and it turns out that it is possible to meet your criteria (all different partners, and on court with each other person at least once). Here is a possible schedule:
(2 7 v 12 17) ( 4 3 v 1 5) (10 8 v 9 6) (11 13 v 14 15) (20 18 v 19 16)
(5 10 v 15 20) ( 1 2 v 6 7) (11 12 v 16 17) ( 3 8 v 13 18) ( 4 9 v 14 19)
(1 7 v 13 19) ( 2 8 v 14 20) ( 3 9 v 15 16) ( 4 10 v 11 17) ( 5 6 v 12 18)
(3 10 v 12 19) ( 4 6 v 13 20) ( 5 7 v 14 16) ( 1 8 v 15 17) ( 2 9 v 11 18)
(3 6 v 14 17) ( 4 7 v 15 18) ( 5 8 v 11 19) ( 1 9 v 12 20) ( 2 10 v 13 16)
(1 10 v 14 18) ( 2 6 v 15 19) ( 3 7 v 11 20) ( 4 8 v 12 16) ( 5 9 v 13 17)
(1 6 v 11 16) ( 2 3 v 4 5) ( 7 8 v 9 10) (15 13 v 14 12) (17 18 v 19 20)
The schedule has 4 repeated opponent pairs (3 5), (8 9), (13 14), (18 19) which all occur in the 1st & 7th rounds. Finally note that the partnerships are all formed from within two groups, players 1-10, and players 11-20.
[comment added later: 20 players is doubly interesting since it is not possible to meet the "on court with each other at least once" criterion with 19 players. For 19 players there must be at least two players who have only 5 games (2 byes), and who must have at least three other players who they never meet.]
