"Where is a big department store?"
I also wrote that (and was not correct), but I had a feeling it should be "doko ni desu ka". Can someone explain why "kudasai"?
Because desu is not a straight translation of "is". It's more a polite ending a lot of the time. Doko desu ka or doko ni arimasu ka, but not doko ni desu ka.
Please don't rely solely on duo for learning any language. It's a good start and supplement, but if you really want to learn, you would be better off investing in a good course. Tae kim has a great guide you can buy, or download from his site for free. There's way too much duo misses out on (like how to use desu and da for example!) especially if you use the app (still can't believe no tips and notes yet!)
I could be completely wrong here, so someone can comment on if this is right, but the use of ___はどこですか sounds more like you know the object in question already exists in area and are asking where it is. Meanwhile, this question is asking if there is one, and asking where it is at the same time. I'm inferring this based on the fact that ありますか is like asking "Is there?"
This English question doesn't sound very natural either. I personally would have split it into two phrases "Is there a big department store here?" "Where is it?"
You have that exactly right. I missed it the first time. It is asking where is a department store if it exists hence we have to use the ある。It is not asking where is the department store.
大きな is more subjective, like 大きなもんだい where as 大きい is more literal, and describes more physical things like a store. Same as おもたい can be slightly more subjective than おもい (heavy).