Like the articles ("the", "a", etc.), the interrogative pronoun "which" (welch-) has to agree in gender/number (masculine, feminine, neuter OR plural) and case (nominative, accusative, dative, genitive) with the noun it refers to. Here, "which" refers to the noun "drink" (German: Getränk). "Getränk" is:
2) accusative (it's the direct object in this sentence, which becomes perhaps more apparent if you answer the question: I like this drink (= direct object). Almost all direct objects are in the accusative case in German.)
For neuter accusative, you have to use "welches". "Welchen" would e.g. be used for masculine accusative. Ex: "Apfel" (apple) is masculine, so you would say: "Welchen Apfel magst du?" (Which apple do you like?).
See this table:
You're right that both the interrogative and the possessive pronouns have to be nominative in your sentence. They also both have to be masculine, because "Hund" is masculine.
"Welcher" is nominative masculine and correct.
However, "dein" is not quite right here. "Dein" is nominative masculine, but it means "your", not "yours" (e.g. Dein Hund ist schön = Your dog is beautiful). So your sentence means "Which dog is your?". "Dein" in the sense of "yours" can sometimes be found in archaic/poetic texts, but the modern nominative masculine form of "yours" is "deiner": "Welcher Hund ist deiner?" (Which dog is yours?). See here, second table from the top: http://www.canoo.net/inflection/dein:Pron:Poss:2nd:SG
Generally, people often say "Welcher Hund gehört dir?" (Which dog belongs to you?) instead of "Welcher Hund ist deiner?" (Which dog is yours?).
IS this the question one would ask when offering someone a drink? If so, then "What (or which) drink would you like?" should be a correct English translation (even though it isn't strictly present tense).
The exact present tense, "What (or which) drink do you like?" is asking about someone's general likes. It is the question you would ask when grocery shopping.
Now we need a native German speaker to tell us exactly when "Welches Getränk magst du?" would be used.
I think then you'd use 'Welches Getränk moechtest du?' (I've used 'oe' here as I cannot find an umlaut :))... So it's more which drink DO you like vs. which drink WOULD you like (as in, the first one you could imagine that you're tasting two different drinks and your friend asks you which one do you like).