Thats the thing - "Are you taking this dog?" may also refer to future plan (according to https://englishlive.ef.com/blog/language-lab/english-grammar-help-using-ing-for-future-plans/ ) and still polish sentece is in Present Tense, not future as the translation would suggest. Thats why I am confused.
It may be because I am a Czech and it would be kind of possible in czech to use in this case both "bereš" (bierzesz) and "vezmeš" (weźmiesz).
Another thing is that as there is missing context it is harder to determine right tense as it is pretty common with translating slavic languages to germanic languages namely english.
I agree with making Future Tense obvious by using "will" or "going to". But there are times, when you use -ing form of the verb to express Future Tense. Every language has its own matters that are hard to learn by foreigners.
Sorry for nitpicking. So to preserve the topic and to return to your first reply:
"Brać" - used for Present Tense and Past Tense only (?) Future Tense only when using Future Tense of "być" (?) "Wziąć" - used for Future Tense and Past Tense only (?)
So in this case the sentence is in the Present Tense, therefore Present Continuous is the option. Is it possible to use Present Simple and translate it to "Do you take this dog?" ?
It is Accusative, but in Accusative, for masculine nouns, it matters whether the noun is animate or inanimate (this is the only grammatical 'situation' where it matters).
For inanimate nouns, the Accusative form is identical to the Nominative form, so for example "Do you see this table?" is "Widzisz ten stół?".
For animate nouns, the Accusative form is identical to the Genitive one, and this is what we have here.