Not a native speaker, so take with a grain of salt unless someone verifies:

But, I think it's because of passive action in the Russian sentence. It's not about her hurting her hands or injuring her hands by her own action - her arms hurt on their own, through no fault of hers.

I think you'd have to use the verb form ранить (to wound/injure). This sentence just means that her hands aren't hurting/sore/in pain.

Yes, actually it can. But this sentence would sound a little bit less natural.

In this sentence you put accent to 'никогда' and it sounds like you are little be angry because of this fact:)

I don't think they do, the hand is just part of the lower arm? Q)Where are your hands? A)At the end of my руки

Genitive. Nouns ending in а end in ы, but in this case it is и, because you cannot write я or ы after the "seven consonants" (к, г, ш, щ, ж, ч, х)

Although the meaning is pretty much the same, I think the grammatical structure is just too far off.

The subject is 'her hands', not 'she', and the hands aren't sore (adjective), but actively hurt (verb). Or a negation of that, in the exercise.

I also thought "She never has arm pain". The "У неё", always makes me think "She has".

Yes

I'd expect that in a sort of narrative situation.

—Why didn't she go to the doctor earlier?

—Well, her arms never hurt.