Not when it starts the question, in some other contexts. Like "Zapytał czy jesteś chłopcem czy dziewczynką" = He asked, whether you are a girl or a boy.
It sounds like it's similar in meaning to Ukrainian чи or Russian ли, no? Kind of like a "whether ... or ..." construction, except that it can be used for singular subjects as well.
As far as I know, ли isn't that common, at least it doesn't seem common in the Duolingo course ;) "czy" is a basic word, although well, it's not obligatory.
As a 'surprised question'? If you omit 'czy', it may be understood both as a real question and a 'surprised question'. Depending on the context and the intonation.
It's not really a surprised question, it's more of an assumption question, or asking someone to confirm what they just said or indicated.
e.g. You see your mother, she takes a look at your shirt and makes a face, you are more likely to say, "You don't like my shirt?" than "Do you dislike my shirt?"
So in Polish that would just be minus the czy, so like a statement, but with question intonation?
OK, that was an oversimplification from my side, sure, there are other possibilities than being surprised :)
But the rest stays: you just omit 'czy' (because that indicates a real yes/no question), and use a question-like intonation.
What is the etymology of the word czy? Has it any roots with other slavic languages .
I don't know how this word started being used for starting yes-no questions. But I've heard that some Slavic languages close geographically to Poland have similar words for that (like in Kashubian or Ukrainian).