Correct. Slavic languages do not use articles but English does. If you translated a book from a Slavic language into English, do you think that you would not use any articles in it? Just because it is not on the Czech side does not mean it cannot be on the English side as it might be a more natural way to express the same thing. THAT is accepted as well though.
"To be seeing a dog" means something very different.
"do you see the dog?" is acceptable and correct in english. "you see the dog?" is incorrect both in grammar and unnatural in most all cases
Definitely "Do you see the dog"? is grammatically correct in English. However, at least in the US, you also would definitely hear "You see the dog?" in everyday speech. So perhaps DL accepts it as an alternative translation.