There will be few combinations of the number of players and rounds that lead to a tournament where everyone gets the same number of games (and byes). I counted only 35 possibilities. A lot of these will be where the number of players is 20 and everything is easier because there are no byes. In particular 20 players in 19 rounds is the whist design that you can find by clicking on the 'schedules' link near the top of the page. Another balanced solution will be the whist design for 21 players in 21 rounds, and you can find the first round of a cyclic schedule on this web page (http://www.durangobill.com/BridgeCyclicSolutions.html).
Very few of the other possibilities will work out nicely, and I fear you have no option other than writing or finding software to optimise the properties that you have outlined above - I don't see that there is any easy way here. There is software out there that may help if you try an internet search for tennis doubles software.
For me, the most interesting of the 35 possibilities is 25 players and 15 rounds. It is possible to have a tournament where every player has 12 games and 3 byes, with 12 different partners and oppose every other player exactly once. For example:
Table 1 Table 2 Table 3 Table 4 Table 5 Byes
(23 8 v 14 1) (12 3 v 17 13) (22 24 v 10 6) (11 9 v 20 18) ( 4 25 v 2 16) (21 7 5 15 19)
(24 9 v 15 2) (13 4 v 18 14) (23 25 v 6 7) (12 10 v 16 19) ( 5 21 v 3 17) (22 8 1 11 20)
(25 10 v 11 3) (14 5 v 19 15) (24 21 v 7 8) (13 6 v 17 20) ( 1 22 v 4 18) (23 9 2 12 16)
(21 6 v 12 4) (15 1 v 20 11) (25 22 v 8 9) (14 7 v 18 16) ( 2 23 v 5 19) (24 10 3 13 17)
(22 7 v 13 5) (11 2 v 16 12) (21 23 v 9 10) (15 8 v 19 17) ( 3 24 v 1 20) (25 6 4 14 18)
(15 6 v 1 7) (10 3 v 2 14) ( 5 23 v 12 13) (18 22 v 16 17) (24 19 v 9 21) ( 4 25 11 20 8)
(11 7 v 2 8) ( 6 4 v 3 15) ( 1 24 v 13 14) (19 23 v 17 18) (25 20 v 10 22) ( 5 21 12 16 9)
(12 8 v 3 9) ( 7 5 v 4 11) ( 2 25 v 14 15) (20 24 v 18 19) (21 16 v 6 23) ( 1 22 13 17 10)
(13 9 v 4 10) ( 8 1 v 5 12) ( 3 21 v 15 11) (16 25 v 19 20) (22 17 v 7 24) ( 2 23 14 18 6)
(14 10 v 5 6) ( 9 2 v 1 13) ( 4 22 v 11 12) (17 21 v 20 16) (23 18 v 8 25) ( 3 24 15 19 7)
(10 20 v 8 4) (13 11 v 6 19) (24 15 v 23 12) (22 16 v 1 3) (17 5 v 25 9) (21 2 14 7 18)
( 6 16 v 9 5) (14 12 v 7 20) (25 11 v 24 13) (23 17 v 2 4) (18 1 v 21 10) (22 3 15 8 19)
( 7 17 v 10 1) (15 13 v 8 16) (21 12 v 25 14) (24 18 v 3 5) (19 2 v 22 6) (23 4 11 9 20)
( 8 18 v 6 2) (11 14 v 9 17) (22 13 v 21 15) (25 19 v 4 1) (20 3 v 23 7) (24 5 12 10 16)
( 9 19 v 7 3) (12 15 v 10 18) (23 14 v 22 11) (21 20 v 5 2) (16 4 v 24 8) (25 1 13 6 17)
Hope that helps.