Learn French: Elle n'a jamais abandonné son projet. - She never abandoned her project.

In the Passé Composé, short negative words like jamais, rien, and plus wrap around the helping verb (a), not the main action verb. So it is n'a jamais followed by the past participle abandonné.

Since she 'never' did it, shouldn't this be in the Imparfait to describe an ongoing state in the past?

Even though 'never' sounds like a continuous state, the Passé Composé is used here because it states a completed fact about a specific period. The Imparfait would be used if you were describing a repeated past habit, like 'she never used to abandon projects'.

Do I pronounce the 's' at the end of jamais?

Yes, you make a liaison here. Because jamais ends in an 's' and abandonné starts with a vowel, you pronounce the 's' as a 'z' sound connecting the two words.

Does projet only refer to a work or school project?

Projet often means a literal project, but it is also the standard French word for a plan or intention. For example, asking about someone's projets pour le week-end means asking about their weekend plans!