Usually over mating causes prolapsed penis especially in drakes. so I assume in ganders too.
The farting, though... Have your ganders/ducks farted out their own penises after they've been replaced after a prolapse? Or anyone else you've heard of? If so, perhaps the penis was replaced into the incorrect internal position so prevented them eliminating feces properly and the pressure built up until it came out noisily. (?) I don't know why they would start making loud farts in this case, normally I'd assume they've got a buildup of gas for some reason, in the gut or bowels.
I have heard of various prolapse cases but not the loud farting as a symptom. :/
I would have a different stance on the subject of overmating partly because my stance on overmating in chickens is also different, I believe the general physical and social health statuses of both male and female are undervalued as far as a contributing factor in both cases. I would probably view it in most cases as an animal in subpar health overdoing something a male in prime health wouldn't injure himself from, or perhaps an animal with a genetic predisposition to prolapse, or one on a poor diet causing the mucosal membranes, connective tissues etc to lose flexibility, since the latter is one of the greatest factors in, if not
the greatest factor involved in female fowl prolapsing. Lack of raw oils in the diet specifically.
It's also possible it's a mental issue, not knowing when to stop and rest, I guess that could be termed 'overmating'. But then you'd expect most young male fowl to suffer from it, potential contributing factors aside.
It'd be interesting to see what the records for a flock known to suffer penile prolapse look like, I'd bet there's probably a familial leaning towards it. That'd be a pretty serious issue for a wild bird after all, it's probably a problem that's only reached a common status under human intervention. Even in the wild, geese and especially ducks are quite often prolific maters.
I've never had 'overmating' issues with my chooks because I don't keep roosters who are abusive to hens, and I feed the hens kelp etc to ensure their feathers are strong, so the many people simply stating 'overmating' is due to roosters not having enough hens just baffles me. Especially because roosters with more hens than they can keep 'covered' still tend to have a favorite they abuse. Some of my earlier flock included Isabrowns, and older hens from various places who had terrible feathers to start with, and overmating was not an issue with them either because the roosters were gentle enough and intelligent enough to back off if a hen refused them. It's not about the hens, it's about the roosters, in my experience. Sometimes my ratio of males to females is 50:50 and I've never had 'overmating' damage to the hens despite the frequent mating, barring a few cases of accidental spur scrapes.
If the gander was trying to mate unwilling geese, though, I could imagine some potentially damaging issues occurring.
Anyway, best wishes.