Are men and women permitted to play volleyball against each other, with each team only one gender?
Does it make a difference if they are separated by an opaque wall or a regular net?
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As per the Shulchan Aruch O.C. 21:1, a man is not permitted to be playful with women whose relationship with him is forbidden by punishment of Kareth. This includes married women, (some) relatives and women who have the status of Nidah.
Though I can't quote a source, I think we can safely assume this does not apply to relatives for whom we are not concerned for their sexual relationship (e.g. parent-to-child, siblings, etc.)
Are you somehow comparing volleyball to the partition between the men's and women's seating in a synagogue?
Outside the synagogue, the only rules on contact between genders are:
Especially with regards to that last point, different communities will have wildly different standards as to what's called "reasonable sense." (Or "we adopt blanket rules to avoid problems").