Don't hold me to this but I believe that the object pronouns have to come after the verb in the imperative mood.
In this sentence, could "ei" be "they"? Could it mean something like "You answer me, they do not"?
No. 'Ei' for 'they' is used only for the nominative case; that is, when the 'ei' is performing the action. In the example, the 'ei' is /receiving/ the action (i.e. answering TO...) and this forces usage of the dative/genitive pronoun.
I'm not sure that really answers my question. What I'm asking is precisely whether "ei" could be nominative case "they" rather than dative/genitive case "her" in this sentence.
Is there anything about the sentence that forces "ei" to be read as dative/genitive rather than as nominative?
I actually think it did answer the question because of what nominative, dative, and genitive are.
You asked if anything forces this --
Quote impvam: "In the example, the 'ei' is /receiving/ the action (i.e. answering TO...) and this forces usage of the dative/genitive pronoun."
But could we read it with 'ei' /doing/ the action as nominative, rather than /receiving/ the action as dative/genitive? How do we know, from the sentence, that 'ei' is receiving the action? If you say, 'because it's dative/genitive', then the logic becomes circular.
Is there anything that prevents us from reading the sentence as "You answer to me, they do not"?
A native speaker should answer this, but I guess, since the subject "tu" is missing in the first part, the elliptic "nu ei" will not refer to the subject (nominative) but rather to the object (dative) which is stressed here (îmi + mie).
Tu-mi răspunzi, nu ei.
is probably what you have in mind (nu ei = not they) - but I am just learning this myself...
ei nu is probably the version where ei performs the action.
nu ei = not to her
ei nu = they do not
Non-native, just guessing since nobody seems to catch what you mean.
necronudist85's answer addresses what I was asking about and would make sense - can a native speaker confirm if it's correct?