The words are connected. We have many Kazakh (turk) words, because of the Kun (Cuman) settlers, and we have many old and modern turk words with other origines. We had lived together with turk tribes, and later in the modern times the Osman empire occupied our country.
Hölgyeim és uraim, that is correct.
That is literally "my ladies and my gentlemen".
And there is one more word: "asszony". I would say it is closest to "ma'am" in English. It can be used with adult women who are probably married. It is most commonly used in the possessive: "asszonyom" - "my ma'am".
And a similar term for young women is "kisasszony". The usage of these words is a bit formal. But anyway,
"Hölgyem"/"Asszonyom" - can be used similar to "ma'am" in English or "señora" in Spanish, for example.
"Kisasszony" - can be used similar to "Miss" in English" or "Señorita" in Spanish. So:
Excuse me, ma'am - Elnézést, hölgyem/asszonyom
Excuse me, Miss - Elnézést, kisasszony
Nincs can only replace van to counter positive sentences. And van only appears in sentences (in 3rd person singular and plural but in plural it's vannak/nincsenek) that indicate a place, time or situations not description of treats in some way (using adjectives or nouns instead of adverbials). So in this sentence the only correct negative word is "nem".
If the sentence was something like: "The woman is not outside." then you could use "nincs." "A nő nincs kint."
Of course there is. https://www.duolingo.com/skill/hu/Basic-1/tips-and-notes read this.
for "assumed": I think everyone is free to make up theories about a language as long as they help; I'm not sure thinking of it as something "assumed" helps the most. This is the only verb that can take adjectives and nouns and actually it doesn't describe an action in those cases, it just links concepts to each other. One may say it's hardly a verb at all and the actually strange part is when it's not omitted...