With verbs of eating/drinking and knowing/learning, a reflexive form can (and is very frequently) used. It implies "complete" or "all". Yo me tomé un café. = I drank the (whole) cup of coffee. Yo me aprendí el poema. = I learned the poem (by heart). In speech, I think, it is the more common form.
No, it's commonly used, but it is not actually correct. Drink, Drank, Drunk are the tenses of "to drink" in English. Drink is present, drank is simple past and drunk is the past participle. When you use "have" you need the past participle after it. I almost never hear this used correctly in English, but it definitely should be "I have drunk" and NOT "I have drank".
I was reading a biography about Eva Peron. She refused coffee when the waiter offered it to her, and then put a political enemy in his place by making HIM fetch the coffee. From the book:
"Pardon, Doctor, now I really want a coffee," she told the statesman in the tones a B-girl might use to an uncertain swain. "Come on, little rabbit, fetch me a coffee."
According to SpanishDict (https://www.spanishdict.com/translate/tomar), there are several senses of tomar/tomarse that are relevant to this question: 2b, 3, and 7 of tomar, and 10 of tomarse (technically as pronomial, rather than reflexive, but I'm not clear on that distinction.
Duolingo's insertion of me into its suggested answer is mainly confusing, since it's optional to use sense 10. Duo does this a lot, that is, gives an answer that's confusingly far away from one of my almost right answers (here I omitted ya, of all silly mistakes). That did get me here into the discussion and into SpanishDict, which wound up being beneficial, but I don't think intentionally confusing students away from the focus topic is good pedagogy.
Would it be wrong to say "I had already taken my coffee"? I know that there's no actual "my" in that sentence, but many times I've seen the possesive pronoun implied in Spanish in sentences such as "Me pongo los zapatos"/"I wear my shoes" or "Me lavo las manos"/"I was my hands".