Learn Croatian: Je li tebi uopće stalo do te sitnice? - Do you even care about that trifle?

This sentence uses the impersonal idiom stalo mi je do (I care about / it matters to me). In this construction, the person who cares is the indirect object and must be in the Dative case. So, the concept of "you care" is expressed essentially as "it matters to you" (tebi).

Why is the verb stalo in the neuter form?

Because this is an impersonal expression, there is no active subject (like a specific "he" or "she") performing the action. Instead, "it" is the implied grammatical subject, which forces the past participle stalo to take the default neuter singular form.

How does the do + Genitive part work in this expression?

The idiom stalo mi je is always followed by the preposition do and the Genitive case to indicate exactly what it is that you care about. Here, the phrase ta sitnica (that trifle/little thing) changes into its Genitive form, te sitnice.

Could we use the short dative pronoun ti instead of the full form tebi here?

Yes, but it would change the emphasis and the word order. Using the full form tebi puts strong emphasis on the person: "Do you even care?". If you used the short, unstressed clitic ti, it would have to jump to the second position in the sentence, right after the question particle: Je li ti uopće stalo do te sitnice?