Your original question unfortunately contained more than one question, and so according to enotes regulations I have had to edit it down to one question, focussing on why the narrator killed his wife.
Crucial to understanding this story is understanding the influence of the black cat on the narrator and how what he does to it and then the continual haunting of the black cat drives him ever madder and madder. The guilt he feels at having killed the first black cat, and then the arrival of the second cat that always haunts him drives him over the edge into insanity:
Beneath the pressure of torments such as these, the feeble remnant of the good within me succumbed. Evil thoughts became my sole intimates - the darkest and most evil of thoughts. The moodiness of my usual temper increased to hatred of all things and of all mankind; while, from the sudden, frequent and ungovernable outbursts of a fury to which I now blindly abandoned myself, my uncomplaining wife, alas! was the most usual and the most patient of sufferers.
It is in the immediate paragraph following this that the narrator and his wife go down to the cellar. So angered by the presence of the black cat ever following him, the narrator tried to "aim a blow" at the cat with an axe. His wife stops him, and, in a moment of sheer madness and anger, he brains her.
Thus the narrator kills his wife because of the madness into which he has been driven thanks to the black cat and its continual, haunting presence around him, never giving him a moments peace.