Learn Romanian: Cică șobolanul a intrat prin pivniță. - They say the rat entered through the cellar.

No, it acts as an adverb. It is an informal, very common spoken word that means supposedly, allegedly, or word has it. It actually comes from shrinking the phrase se zice că (it is said that).

Why do we use prin instead of în (into) here?

While în pivniță would mean into the cellar, using prin pivniță means the rat used the cellar as its entry route (through the cellar). It highlights the pathway the animal took rather than just its final destination.

The English translation says 'the cellar', so why isn't it pivnița with the definite article at the end?

In Romanian, most simple prepositions like prin, în, and pe require the noun that follows to be in its bare, indefinite form, even when we translate it with 'the' in English. You only add the definite article if that noun is followed by a modifier, such as an adjective.

Is șobolan a general word for any small rodent?

No, șobolan specifically means rat. If you want to say mouse, the word is șoarece. Learners sometimes mix these up, but they are completely distinct words in Romanian.