There is bad news, as I don't believe that that there will be any good solutions to your scheduling problem if you can only play about 5 rounds. Simply counting the number of other players that a particular player will encounter, means that the with-everyone-else-once criterion can only be met with at least 8 rounds, while the against-everyone-else-once criterion needs at least 6 rounds. In reality the constraint of having everyone play exactly once per round will mean that you need much more than 8 rounds to achieve your aim. The 6 round schedule below may be an acceptable (poor) compromise:
(22 2 3 10 v 4 16 11 6) (8 13 9 17 v 12 18 24 1) (15 19 20 14 v 23 21 7 5)
(23 3 4 11 v 5 17 12 1) (9 14 10 18 v 7 13 19 2) (16 20 21 15 v 24 22 8 6)
(24 4 5 12 v 6 18 7 2) (10 15 11 13 v 8 14 20 3) (17 21 22 16 v 19 23 9 1)
(19 5 6 7 v 1 13 8 3) (11 16 12 14 v 9 15 21 4) (18 22 23 17 v 20 24 10 2)
(20 6 1 8 v 2 14 9 4) (12 17 7 15 v 10 16 22 5) (13 23 24 18 v 21 19 11 3)
(21 1 2 9 v 3 15 10 5) (7 18 8 16 v 11 17 23 6) (14 24 19 13 v 22 20 12 4)
no pair of players is on the same team together more than twice, no pair of players opposes more than twice. The three pairs (13 16) (14 17) & (15 18) never meet either on the same team or as opponents. There are 12 pairs such as (1 9) who play together twice and oppose twice.
I think you would need 23 rounds to have any chance of finding a completely balanced solution.
