As a professor, I can say, Duolingo is not teaching anything, because this app doesn't explain anything. This app has exercises and grades them. People try to learn guessing, watching, thinking, analyzing, researching, trying and reading what other users explain. People will learn according to the effort and time. If you dedicate time and effort you will learn but as I say, there's no explanation.
You're right. Correct English usage is either/or and neither/nor. The above users are thinking of colloquial English for a "she or I" type situation. You would say: "Either she or I should go" or "Neither she nor I should go." (Colloquially, someone might say, "Neither she or I should go." But that's not typically where the use of "nor" is important.)
Think of these sentences:
"Neither of us saw the dragon, nor did we hear it."
No one would say, "Neither of us saw the dragon, or did we hear it." (Sounds weird)
This Russian translation just isn't a compelling sentence example in English for a "neither/nor" usage. We'd say, "Neither of us has a dog," or "I don't have a dog, and neither does she." But the Russian might be exactly what you'd say in Russian.
"Neither the girl or I" should be fine as another translation, unless you're translating at a rather formal level (in which case nor is a better choice). Usually it's "neither ... nor", but "neither ... or" is not wrong. Merriam-Webster has this usage note: """ Usage Discussion of neither
Although use with or is neither archaic nor wrong, neither is usually followed by nor. A few commentators think that neither must be limited in reference to two, but reference to more than two has been quite common since the 17th century <rigid enforcement of antique decorum will help neither language, literature, nor literati — James Sledd>. """ If you have access to the Oxford English dictionary, they provide some examples of "neither ... or" and caution that "Numerous grammarians from the mid 18th cent. onwards criticize the use of or rather than nor as ungrammatical and improper."
Well, according to Wiktionary, девушка is a girl from "the age of puberty to adulthood," while девочка is a girl "before the age of puberty". In English, we would probably refer to them as girl and little girl. There is even another word, девочка-подросток, that means a "girl at the age of puberty". Thanks for asking, I looked it up and learned two new words! :)
When using нет in these sentences, you conjugate the noun that you don't have (usually the thing after нет) in the genitive.
Open the declension in https://en.wiktionary.org/wiki/%D1%81%D0%BE%D0%B1%D0%B0%D0%BA%D0%B0#Noun for собака, and you find the singular genitive is собаки and the plural genitive is собак.
Thus, to say "Neither the girl nor I have dogs." You'd write "Ни у девочки, ни у меня нет собак"