can you ever say אין instead of לא? Or is it only "there is no(t)"
You could say אינה/איננה.
החולָה שלנו לא שוֹתָה חלב או החתול שלנו לא שוֹתֶה חלב
If you use the feminine verb form you also have to use the word for a female cat: הַחֲתוּלָה שׁוֹתָה
Does the sentence need to say החתול, as opposed to just חתול ?
yes. in hebrew we say "the cat that's his" instead of just "his cat". I think.
For close family members you do not need to add ה. Mum, Dad, husband, wife, grandmother, grandfather...I am not sure if a dog counts as a close family member...
What is the difference between the first answer and the second one? Do the diacritical marking make a big difference?
The forms שׁוֹתֶה he drinks and שׁוֹתָה she drinks look the same without niqqud.
ha-khatul shelanu lo shoteh khalav