Learn Italian: Sono esausto; li sto guardando da stamattina. - I am exhausted; I have been watching them since this morning.

Yes! With the stare + gerund construction, you have two choices for object pronouns like li (them). You can place it right before the conjugated verb (li sto guardando) or attach it directly to the end of the gerund (sto guardandoli). Both mean exactly the same thing.

Why does the English translation say I have been watching instead of I am watching?

This is because of da stamattina (since this morning). In English, we use the present perfect continuous for an action that started in the past and is still happening. In Italian, you cannot do this; instead, you use the present continuous (sto guardando) or the simple present (guardo) along with da to express an ongoing action.

Is stamattina a single word?

Yes, it is a very common contraction of questa mattina (this morning). You can use either one, but stamattina is what you will hear most often in spoken Italian. You will also see this pattern with stasera for questa sera (this evening).

If a woman were speaking, would the sentence change?

Yes, but only the first part! A woman would say Sono esausta, changing the adjective ending to match her gender. The rest of the sentence stays exactly the same, since li refers to the masculine plural things being watched, not the speaker.