Learn Russian: У тебя на щеке синяк, кто разбил тебе губу? - You have a bruise on your cheek, who busted your lip?

This is the Dative of inalienable possession. When an action directly affects a body part, Russian prefers to use the Dative case for the person affected (тебе) rather than a possessive adjective (твою). It emphasizes that the action happened to you and caused you detriment or harm, rather than just acting on an object you happen to own.

If body parts take the Dative in these situations, why does the first half use у тебя на щеке instead of a Dative form?

The Dative of detriment requires an action that affects the person, usually expressed by a verb like разбил (busted). The first clause is a simple statement of existence or location (having a bruise on your cheek). Without an active verb causing an effect, Russian defaults to the standard у + Genitive construction to show possession.

Разбить usually means to shatter or to break (like a vase). Is it really the standard verb for a busted lip?

Yes, it is! While it does mean to shatter or break into pieces, разбить is the standard and most natural verb to use when talking about breaking the skin through a blunt force injury. You will commonly hear разбить губу (bust a lip), разбить нос (bloody a nose), or разбить бровь (split an eyebrow).

Could I use this same Dative of detriment structure if someone broke something I own, rather than a body part?