None of them is too common in Czech. We normally ask "Máš (nějaké(ho)) sourozence?" (either plural or singular).
Other options are less common. Máš bratra nebo sestru? bráchu nebo ségru?
The corpus clearly shows we use the singular much more often than the plural.
A direct comparison: "bratra nebo sestru" 86 hits, "sestru nebo bratra" 21 hits. Regardless of context, often with some different verb.
"nějaké sourozence" 92 hits, often in connection with "máte" or "máš", "nějakého sourozence" 19 hits.
It would always be "Do you have any brothers or sisters?" (with rising intonation at the end which doesn't fall) . . . The reason is that questions without adverbs, beginning with the verbs do or be, are yes/no questions, unless they contain or in which case the invisible witch, err, which causes to be long-answer questions assuming one or the other but not both (with rising intonation on the first, and falling on the second). Adding any (nějaké(ho)) seems to get around this somehow. You can distinguish with intonation if speaking, but the any is necessary in clear writing in English, though you can see it is not so in Czech:
"Máš sestry nebo bratry? Yes, I have a sister and a brother.— Ano, mám sestru a bratra."
"(Which) Do you have(,) brothers or sisters?" gets "(Co) Máte(,) bratři nebo sestry?" It could be "Co máš" instead of course . . .
I am native AmE, and I wouldn't say that it would ALWAYS be "Do you have any brothers or sisters?" But I would agree that the standard English phrasing puts "brothers" before "sisters"and that phrasing with "any" is more often used than phrasing without it. However...
This is a translation exercise, and the Czech sentence which we are to translate to English doesn't include "any" and it places "sisters" before "brothers."