"My uncle does not have any children."
Translation:Mój wujek nie ma żadnych dzieci.
"Żaden" is not a adverb, but a negation pronoun . It is more like English "none" than like "any", because "any" can translate to "jakiś/jakieś". "Żaden" is used only in negation.
And yes, we use double negation in Polish: "nie ma żadnych". You can also imagine a correct sentence with more negations: "Nigdy nie ma żadnych niejasności" - [There are never ever any uncertainties].
„Dziecko” is singular, the plural is „dzieci”. „Źaden” is not a word, it's „żaden”. And its non-masculine plural genitive form is „żadnych”. „Żadne” is non-masculine plural nominative (and technically accusative too, but I can't think of any example where it would ever be used in accusative).