Learn Croatian: Dao mu ga je jučer. - He gave it to him yesterday.

Usually, auxiliary verbs do come first. However, the 3rd-person singular auxiliary je is a strict exception in Croatian! It always drops to the very end of the clitic chain. That is why it appears after the pronoun clitics here.

Is there a specific rule for the order of mu and ga?

Yes! When you have multiple pronoun clitics in a row, the dative pronoun always comes before the accusative pronoun. Here, mu (to him) is dative and ga (it) is accusative, so the strict order must be mu ga.

Since the subject 'he' is dropped, could we start the sentence with one of the short pronouns?

No, you can never start a sentence with enclitics (unstressed words) like mu, ga, or je. They need a stressed word to lean on. Because the subject pronoun (on) was dropped, the main verb dao takes the first position to act as their anchor.

The English translation says 'gave it'. How do we know ga means 'it' and not 'him'?

Grammatically, ga can mean either 'him' or 'it' because it is the accusative form for both masculine and neuter nouns. You have to rely on context. Here, you wouldn't typically give 'him' to 'him', so it makes the most sense as 'it' (referring to a previously mentioned masculine or neuter object).