But letting them happen can cause problems. WOuld they allow gay or lesbian marriages or force only hetero ones?
The Next Generation was the first North American television show to have a homosexual kiss on screen. I don't think homosexual marriages would be an issue.
A lot of the in game marriages that I have known have been between real life couples. Others have been joke/sham marriages just to do it. Only a very few have fallen into that "crazy fantasy life overload" type of marriage. Also, whether or not they have marriages will hardly stop people from "hooking up" in game and becoming one of those crazy fantasy life overload people.
WOuld they allow gay or lesbian marriages or force only hetero ones?
Considering the setting, the Star Trek universe (where even adaptive programming gets equal rights to biological lifeforms), its safe to say they would obviously allow gay or lesbian marriages. The god-fearing zealots who oppose this shouldn't be playing this game in the first place, as the entire principle of Star Trek undermines the dogma's of their religion(s).
Personally, I can't really see the added benefit of game-supported ingame marriages. If someone is into these things its very doable to make it work in a player-run roleplay event. One could always buy a namechange with cryptic points and take the same last name, and as for rings, well some cosmetic jewelry would be nice regardless of marriages.