Just to be clear (and hopefully clear up some of the confusion around this), the "o" does not direct the pronoun meaning of the sentence and using the articles before possessive adjectives and possessive pronouns (ex: a minha [mae], o meu [pai]) is always appropriate in any kind of Portuguese (including the Brazilian sort). And, in fact, in Portuguese outside of Brazil (Europe, Asia, Africa which together get referred to as "EU PT") it is pretty much the only way, except with single family members (as in the examples above) where it is okay (but still optional) to drop the article when the noun is present (it is a biblical thing then).
However, in Brazil (in some parts more than some of the others) the articles are largely optional with the possessive adjectives such as, my car, his house = o meu carro/meu carro, a sua casa/sua casa. But, not so optional with the possessive pronouns – ex: mine, yours, hers, his, theirs = o meu/a minha, o seu/a sua, etc), and the article is always necessary with the construct such as, o carro dela (literally, the car of hers).
In any case, YOU (we) can always use it (to avoid the confusion on when to use it) and YOU (we) should never be wrong on Duo for using it, except when it needs to be reported for Duo not accepting the article (usually it is in the more advanced exercises where this has been an issue).
You are correct.
Then it would be, "A gata" instead of, "O gato" in the sentence if the cat is drinking milk that belongs to the cat. Unless the male cat is drinking the milk belonging to a female (perhaps the owner's milk). But that is a much more complicated idea (which highlights that context is almost always lacking on Duolingo).
Seu is not masculine in the sense of who it refers to but rather to the noun it attaches to (in this case the masculine leite) so, o seu leite can mean, his milk, her milk, their milk, your milk, its milk; whereas teu leite means only, your milk, and o leite dele (the milk of his) means only, his milk, while o leite dela means, her milk, and o leite deles/delas means, their milk. But I believe all that comes in a later lesson on DL than this sentence is at.