Learn Italian: Marco sta cercando l'auricolare. Lo avrà perso. - Marco is looking for the earphone. He must have lost it.

This is a great example of the "epistemic future" (using the future tense to express a guess or probability). When you use the Future Perfect to talk about a past event, it means you are making a strong assumption that it happened. In English, we naturally express this kind of deduction by saying someone "must have" done something.

What does lo refer to, and why is it placed before the verb?

Lo is a direct object pronoun meaning "it", and it replaces the masculine noun l'auricolare. In Italian, object pronouns almost always go directly before the conjugated verb. Even though this is a compound tense, the pronoun still goes before the auxiliary verb avrà.

Where is the word for "for" in sta cercando l'auricolare?

You don't need one! The Italian verb cercare inherently means "to look for" or "to search for". Adding a preposition like per after the verb is a common trap for English speakers; cercare always connects directly to the object you are looking for.

Could I just say Marco cerca l'auricolare instead of sta cercando?

Yes, you could. The simple present (cerca) can translate to both "looks" and "is looking". However, using the present continuous (stare + the gerund cercando) strongly emphasizes that Marco is deeply involved in the middle of looking for it right at this very moment.