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?
First of all, it's just dative here, no genitive.
Now, to answer your questions:
1 Anything is possible, but if you want to say they instead of her then you'll have another sentence with another meaning.
E.g. "Tu imi raspunzi, ei nu." means "You answer me, they do not".
If you say "tu imi raspunzi mie, ei nu" it means different things, depending on how you read the sentence. If you emphasize "raspunzi" it may have 2 meanings: "You answer me, they do not" and "You answer me, but you do not answer her". If you emphasize "mie", it only means "You answer (to) me, but not (to) her".
You may have noticed "tu imi raspunzi mie, ei nu" is different from "tu imi raspunzi mie, nu ei".
2 To answer you second question, what forces "ei" to be "her" rather than "they" is the structure "mie, nu ei" which means "me, not her". The word "nu" links the words "mie " and "ei" which makes "ei" be the same case as "mie" and also emphasize the word "mie". Now look again at the first example "Tu imi raspunzi, ei nu." The absence of "mie" makes it impossible for "ei" to be "her".
It is complicated, but you asked.
To make it really simple, let's add a few words to our examples so you can understand better what they mean:
"Tu imi raspunzi, ei nu" is in fact "Tu imi raspunzi, ei nu imi raspund."
"Tu imi raspunzi mie, ei nu" can be "Tu imi raspunzi mie, ei nu imi raspund" and "Tu imi raspunzi mie, ei nu ii raspunzi."
There is nothing to be added to "(Tu) Îmi răspunzi mie, nu ei."