Feminine form of Soy nino, bebo leche
Would the feminine form of the top sentence be Soy nina, bebe leche?
Simple, you add/change final "o" of both determiner and substantive/adjective into "a":
Soy una niña, bebo leche
"Bebo" is first person singular (present indicative) for both genders.
No. The verb conjugation form doesn't change according to gender, it changes according to the speaker - I, you, he/she, they, we etc.
So the conjugation for beber, to drink in your example would be:
Yo soy un nino, bebo leche. I am a boy, I drink milk. And
Yo soy una nina, bebo leche. I am a girl, I drink milk.