Learn Italian: Piegato il gancio, la donna ha infilato il ciondolo. - Once the hook was bent, the woman strung the pendant.

This is the "Absolute Past Participle" structure you are currently learning. By using just the past participle (piegato) right before a noun (il gancio), Italian creates a neat shortcut for temporal clauses. It translates to "Once the hook was bent" or "The hook having been bent," removing the need for phrases like dopo che (after).

Does piegato agree with the woman or the hook?

It agrees with il gancio (masculine singular). In these absolute constructions with transitive verbs, the participle acts passively. You treat the object that was acted upon (the hook) as the grammatical subject of that mini-clause, so the participle's ending must change to match its gender and number.

Could I use a gerund here and say Avendo piegato il gancio instead?

Absolutely. Avendo piegato il gancio (Having bent the hook) means the same thing and makes it actively clear that the woman did the bending. The Absolute Past Participle (Piegato il gancio) is just a more concise, slightly more formal alternative that focuses on the completed state of the hook.

I thought infilare meant to put on clothes. Why is it used with a pendant?

Infilare broadly means "to insert," "to slip in," or "to thread." While it is frequently used for slipping on an item of clothing (like infilare le scarpe or a jacket), here it describes threading or sliding the pendant (il ciondolo) onto a necklace chain or string.