If your fish is well cared for then disease should not occur very often. When your fish do become ill you should act immediately. This topic is so broad that I can only give a general outline here. As ominous as these diseases sound you should be aware that most are easily treatable if you recognize the signs and treat your fish quickly. As always Prevention is the best cure.
His gills will flare up.
they use there gills to reproduce
Not neccessarily. Some Bettas do have very dark gills.
Yes, the "red thing" is actually called a membrane. Bettas will flare out their gills and membranes when he is feeling threatened, especially by another Betta fish.
No bettas are fish not mammals. Bettas have a labyrinth organ which enables them to breathe air when there is not enough oxygen in the water to sustain them. Here are some things that separate bettas from mammals Hair. Bettas have no hair or fur Scales. Bettas have scales, mammals do not Ability to breathe through gills. Mammals do not have gills, bettas do Egg laying. Bettas lay eggs, mammals give birth to babies without shells that are able to breathe by themselves
The male will puff up his gills to scare the female by looking bigger. If the male betta looks straight at you when he is flaring, it makes him look bigger. He follows the female to scare it away.
Some bettas just have dark gills, even the white ones.
This depends on the type (and probably species) of fungus. What particular fungus (fungi) are you interested in?
Mainly for territorial and mating purposes. Bettas are very territorial and will attack any fish with bright colors and large fins. Especially other Bettas. Bettas flare when threatened by opening out their fins and extending a 'beard' from under their gills.
a mushroom or fungus
Bettas and other members of the anabantoid family of fish, which includes the gouramis, paradise fish, snakeheads and combtails, all have adapted to life in stagnant, oxygen depleted ponds and pools. In many instances, they cannot get enough oxygen using their gills alone. They have therefore developed an organ called the labyrinth organ, a long, coiling airway that enables fish from this family to breate atmospheric oxygen, thus reducing their dependance on the oxygen held in the water. In many fish of this family, their dependance upon atmoshpheric air is so great that one can if it is prevented from reaching the surface.
It means he hates you. Bettas flare their gills out when they're trying to scare another betta away, or to impress a female with how large he is.