Breakdown of Тебе что-нибудь принести?
тебе
to you (informal)
что-нибудь
anything / something (unspecified)
принести
to bring (perfective)
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Questions & Answers about Тебе что-нибудь принести?
Why is it что-нибудь instead of что-то here?
Что-нибудь means 'anything at all' and is perfectly suited for open questions, offers, and requests. If you used что-то (something), it would imply a specific thing that already exists in your mind but you aren't naming. Since you are asking an open question about what they might want, что-нибудь is the natural choice.
Where is the 'Should I' in the Russian sentence?
Russian often drops pronouns and uses just the infinitive verb to make a suggestion or offer. Saying принести? (to bring) with a questioning intonation is a common, natural way to ask 'Should I bring...?' or 'Do you want me to bring...?' without needing extra words like я (I) or должен (should).
Why does the sentence use Тебе?
Тебе is the Dative case form of ты (you). The verb принести (to bring) takes the Dative case for the person receiving the item (the indirect object). So, you are literally asking 'To bring anything to you?'
Could I use the imperfective verb приносить here?
Not usually for this kind of offer. Принести is the perfective form, focusing on a single, completed action with a result (bringing something this one specific time). You would only use приносить if you were asking about bringing things repeatedly or as a regular habit.