To me (native AmE), "Does an animal exist without a neck?" seems to be asking whether an animal can live if it has no neck. On the other hand, "Does an animal without a neck exist?" is fine, and that is an accepted translation of the Czech sentence.
Are you sure about this word order? Neither Google or ngrams could find anything similar.
To clarify, "Is there an animal without a neck?" and "Does an animal without a neck exist?" are accepted. What is not accepted is the placement of the verb in your suggested sentence.
Isolated, yes. But when "animal" is further determined by "without a neck" then this expression needs to stick to "animal" and cannot be separated from it by the verb (exist).