Learn Italian: Ha sporcato il tappeto del salotto; sto pulendolo. - He dirtied the living room rug; I am cleaning it.

With the progressive tense (stare + gerund), you have two options for object pronouns like lo (it). You can attach them directly to the end of the gerund, making pulendolo, or you can place them before the conjugated verb stare, making lo sto pulendo. Both are perfectly correct and natural.

When attaching lo to the gerund, do I need to drop any letters?

No, you keep the entire gerund intact. You simply take pulendo and add lo to get pulendolo. This is different from infinitives, where you have to drop the final -e before adding a pronoun (for example, pulire becomes pulirlo).

Why does the sentence use del salotto instead of just di salotto?

In Italian, when specifying which room a piece of furniture belongs to, you almost always use the articulated preposition (preposition + definite article). Del is the combination of di + il, so you are literally saying the rug of the living room.

Ha sporcato could mean he, she, or formal you. How do we know it means he here?

Out of context, we actually do not know for sure! Ha sporcato translates to he, she, or formal you dirtied. The translation here uses he as a default example, but depending on the conversation, it could easily mean she dirtied the rug.