Learn Polish: Owca je i śpi. - The sheep eats and sleeps.

Context is everything! While je is indeed the object pronoun for "them" (non-masculine) or "it", here it is functioning as a verb. It is the "he/she/it" form of the verb jeść (to eat). Because we have a subject (owca) and another verb (śpi) connected by i (and), je has to be the action "eats".

In English, "sheep" can be singular or plural. Is owca the same?

No, Polish uses distinct words for singular and plural. Owca is specifically one singular feminine sheep. If we were talking about multiple sheep, the noun would become owce, and the verbs would also have to change to match the plural subject: Owce jedzą i śpią (The sheep eat and sleep).

The dictionary form of "to sleep" is spać. Why does it change so much to become śpi?

The verb spać has an irregular stem change when conjugated. The a disappears, and the consonants soften: the s becomes ś, and the p gets an i added to soften it (pi). It is one of those highly irregular verbs you just have to memorize. For comparison, "I sleep" is śpię, "you sleep" is śpisz, and "he/she/it sleeps" is śpi.