I agree but DL (still) doesn’t accept it here. To my mind, “to think of” and “to think about” are interchangeable in English in most contexts, and certainly in the sentence here.
Incorrect rendering: "Are you thinking of/considering staying in this country?" sounds more like it.
Not a native speaker, but "Do you think of staying in this country?" sounds quite awkward with the tense used.
"Are you thinking about staying in this country?" should be an acceptable answer too.
Yes. When i start a sentence with "Do you think ..." I am probably going to disagree with what comes next. "Do you think a native speaker would say this?". No. I would say "Are you thinking of staying?"