Learn Italian: Con questi tappi, dormo sodo tutta la notte. - With these earplugs, I sleep soundly all night.

Some adjectives in Italian can be used directly as adverbs to describe an action. This happens in specific, highly common pairings. For example, dormire sodo means 'to sleep soundly' and lavorare sodo means 'to work hard'. In these idiomatic verb phrases, you drop the idea of a -mente suffix and just use the plain adjective.

If the speaker is a woman, or if we were talking about multiple people, would sodo change to soda or sodi?

No, it remains sodo. When an adjective is used adverbially (to modify a verb), it becomes invariable. This means it is locked in its masculine singular form. Because sodo is describing how the sleeping happens rather than who is doing the sleeping, it doesn't need to agree with the subject's gender or number.

Is tappi the specific word for 'earplugs'?

Tappo is actually a broad word meaning 'plug', 'cork', or 'cap' (like a bottle cap). However, in contexts involving sleep, noise, or swimming, tappi is widely understood as shorthand for tappi per le orecchie (literally, 'plugs for the ears').

Could I leave out the article and just say tutta notte?