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It seems in this sentence:
Она не верила своему мужу.
"Man" is in Dative case. 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?
In addition, why is this not in Genitive case, seeing as there is negativity/denial?
Lastly, why is "история" in Dative in this case:
Я не поверил ее истории.
Shouldn't it be in Genitive, seeing as it is "possessed" by her?
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.
The relationship in both of your examples is the same, so the case is the same. It doesn't matter that истории is inanimate. (However, in some circumstances animate and inanimate words may change differently in the same grammatical case).
Why Dative? Well, we can invent some mental bridges that can help to remember it, but the point is that this sort of relationship requires Dative in Russian. You might call it 'acting upon' (there is nothing 'possessive' in it), but I'm not sure it's 100% accurate. In school, we learned it was a 'give to' sort of relationship.
"истории", BTW, is not the best example because its Genitive and Dative forms are identical. But compare e.g.
Я не поверил ее совету.
(Genitive would be 'совета').
Without long explanations: верить + Dative.
