"I do not know her, nor do I want to know her."
Translation:Mi ne konas ŝin, nek volas koni ŝin.
Scii is to know facts. You know physics, how to play with a ball, what color the sky is, etc.
Koni is to know a person, be aware of an idea, have a familiarity with a place, relate to an ambition. One konas a tree by its fruit. One scias only what one learns. etc.
You can koni something, but not scii it, and vice versa.
"Si" is the reflexive pronoun—that is, it refers to the "self" of the subject. "Mi ne konas ŝin, nek volas koni sin" translates to "I don't know her, nor do I want to know herself". (It's a little tricky, because it can also strictly translate to "...nor do I want to know myself" because "mi" is the subject but I'll talk about that later. This sentence wouldn't happen in your average E-o writing since it's grammatically incorrect.)
Here's another example of "si" vs "ŝi": "Ŝi dancas kun ŝin" translates to "She dances with her [i.e. a different woman]"; "Ŝi dancas kun sin" translates to "She dances with herself".
"Li ludas kun lian hundon" translates to "He plays with his dog [i.e. the dog owned by another man]"; "Li ludas kun sian hundon" translates to "He plays with his [own] dog". In each case, the reflexive pronoun "si" explicitly refers to the subject.
"Si" isn't used for first- or second-person pronoun uses (mi, vi, ni) because the reflex of the pronoun is obvious. ("Mi ne konas min, nek ne volas koni min"—"I don't know myself, nor do I want to know myself") But in the third person, it is ambiguous whether the subject and object are the same person so the reflexive pronoun "sin" is used.
After this long ramble, the point is: "si" is not appropriate to use in this case since the subject of the verb and the object of the verb are two different people.
Ne is a negative term. It takes whatever is being said and makes it a negative statement. Hence it can mean all of those things which you list, and probably a few more.
For example: Mi konas ŝin is a normal sentence, one we can deem as positive. Toss a ne before the verb and now I don't know her. Toss that ne before one of the pronouns and it can be "Not I knows her (implying someone else does, just not you) or "I know not her" (Implying that you know someone, just not her). Mi iros (I will go) can become mi ne iros (I won't go). Ne respondu = Don't answer. etc.
Strictly, ne negatives the word it precedes, though this distinction is not always observed. Ne tute = not quite (entirely) but tute ne = by no means, not at all.
Ne~ is also used as a prefix analogous to the English "un~", "in~", "ir~", and "non~" f.e. nedeviga = optional, neklara = indistinct, unclear, nemetala = non-metallic, and neregula = irregular.
Mi esperas, ke vi ne plu estas neklara pro ne.
How is it unhelpful? It's Esperanto's way of expressing a negative, like the English "not", "no" and "non-". Of course, English has many other ways of negating ("regular" becomes "irregular", "standard" becomes "non-standard", and so on). It's so much easier in Esperanto, which just uses "ne", either as a separate word, usually meaning "no", or as a prefix to negate the word to which it is joined, as Fred Capp explains in his post. I don't understand what "May the odds be evr in your favor, tribute." means. Sorry!
Duolingo now allows Mi nek konas ŝin, nek volas koni ŝin – literally "I neither know her nor want to know her", but it continues not to allow Mi nek konas nek volas koni ŝin. Is it valid for two verbs to share an object in that manner in Esperanto? Would saying Mi ŝin nek konas nek volas koni be any better?