"She is sleeping without her cat."
Translation:Sie schläft ohne ihre Katze.
Why is it not 'sie Katze'? Isn't it accusativ ?
You wouldn't say "without him cat", would you? You don't need the accusative case personal pronoun "him" here, you need the possessive "his".
And similarly in German: you don't need the personal pronoun sie, you need the possessive ihr -- in this case in the form ihre for feminine accusative.
Confusingly, English uses "her" for both the personal pronoun and the possessive determiner. "I buy her flowers" can be the female version of both "I buy his flowers" and "I buy him flowers".
Why is "I am against her." = "Ich bin gegen sie"?
Because this is personal pronoun "her", not possessive "her". The male version would be "I am against him", not "I am against his".
I am against her. = Ich bin gegen sie.
I am against her book. = Ich bin gegen ihr Buch.
(Male versions: "I am against him" but "I am against his book".)