Learn Italian: Avendolo osservato, il regista corregge la postura e il gesto. - Having observed him, the director corrects the posture and the gesture.

A past gerund is a compound tense made of two words: an auxiliary verb (avendo or essendo) and a past participle (here, osservato). When attaching a pronoun, it always hooks onto the end of the auxiliary verb. It is never attached to the past participle.

Since there is a direct object pronoun (lo), does the past participle (osservato) need to agree with it?

Yes, it does. With past gerunds, the agreement rules are exactly the same as with the passato prossimo. Because the direct object pronoun is the masculine singular lo, the participle ends in -o. If the director had observed a female actor, it would be avendola osservata.

Why does the main clause use the present tense (corregge) when the first part is in the past (avendolo osservato)?

The past gerund (avendo + past participle) is used to show that an action was completely finished before the main action happened. The main action itself can be in any tense. Here, the director observed him first, and now (in the present) he corrects him. If it were the present gerund osservandolo, it would mean he corrects him while observing him.

Why does the word regista end in an 'a' if the article is the masculine il?