Shouldn't it be in Genitive, seeing as it is "possessed" by her?
This is not how genitive works. It marks the possessor without modifying the possessee in any way.
"Man" is in Dative case.
("Husband" actually; "man" is мужчина.)
Is this because he is the one the verb is "acting" upon, i.e. the receiver of the action or is it another rule, governing the "believe" verb?
Actually it's just a specific verb governed in a specific way. Same as in English, you listen to someone, rather than "listen them", for what seems to be no particular reason — while in Russian, the object of слушать is in the accusative, not dative.
There is a tendency for verbs describing states of mental receptiveness — верить, доверять "to trust", внимать "to heed" — to take a dative object, but while it may explain the original reasons, it's not really useful in predicting whether you'll need dative or accusative.
In addition, why is this not in Genitive case, seeing as there is negativity/denial?
The negative genitive only comes into play where the affirmative would use accusative — meaning, additionally, that the point is moot with masculine animate nouns such as муж, since their genitives and accusatives are identical.