Well, all are Dative. "mu" is the neutral option. "jemu" gives emphasis. And "niemu" has to be used after a preposition.
I think I understand how to use mu, but can you explain what a "neutral option" is? So with jemu it always sounds like emphasis? Never natural?
"Daję jej wino, a jemu sok" - 'jemu' is frankly the only option here, because the way this sentence is constructed, the emphasis is necessary - it's contrasted.
"Daję jemu sok" would be correct, but not that natural as the perfectly neutral "Daję mu sok".
You say niemu follows a preposition. Is there an emphasized version of niemu or do you still use jemu? (emphasized meaning placed emphasis)
No, after a preposition you always have to use the n- version of the pronoun (if such a version exists).
With Dative, there aren't many prepositions, but for example "dzięki niemu" (thanks to him) or "przeciw niemu" (against him).
"A juice"? I don't think I've ever heard that expression in English. I'd go with "I am giving him juice."
I am still not getting this defining PL verb use by EN simple-continuous tense use. It reinforces source grammar, and now i am not surprised that translators make so many tense errors :(