"Does everyone in Turkey have a cezve?"
Translation:Türkiye'de herkeste cezve var mı?
we have conjunctions in Turkish and -de is one of them. But they can make variations in sentences. We remember that if you delete -de in the sentence if it still makes sense it writes separate and like "de" but sometimes if the word ends like -t word it changes and become -te and we write it compound with the last word like this sentence: herkeste.
You can't add the possessive suffix without using the genitive elsewhere. You can hide the genitive word in the sentence, though. You can say: Cezvesi var mı? or Sergey'in cezvesi var mı? Even if you omit Sergey'in, it's understood.
But you can't use Sergey in any other declension, such as Sergey'de. That doesn't cope with the possessive.
If you say "cezvesi", then you should use the possessive form of "herkes" and that is "herkesin". e.g. "Does everyone in Turkey have a ball ➡Türkiye'de herkesTE bir top var mı?" But if you say "topU", then you should say "herkesİN (because of topU) and it makes the sentence as "Türkiye'de herkesİN bir topU var mı?"