Learn Croatian: Ona hoće završiti projekt! - She will finish the project!

Because it adds emphasis! The short form će is used for the regular future tense (Ona će završiti...). Using the full form hoće makes it an emphatic future, which gives the meaning of She will finish the project! and shows strong determination.

Since hoće is the full form of htjeti, could this sentence also mean She wants to finish the project?

Yes, absolutely! The full forms of htjeti (hoću, hoćeš, hoće, etc.) are used for both the emphatic future tense and the present tense of the verb meaning to want. In real conversation, you rely on the context and the speaker's tone to know if they mean a strong intention or a desire.

If we drop the pronoun Ona, can we still start the sentence with Hoće?

Yes, you can just say Hoće završiti projekt! Because hoće is an independent full word, it is allowed to start a sentence. If you were using the normal short form će, you could not start the sentence with it, because those short words (clitics) are never allowed in the very first position.

Why did the word projekt stay exactly the same as its dictionary form even though it is the direct object?

Projekt is the direct object here, which means it is in the accusative case. However, for masculine nouns that represent non-living things (inanimate nouns), the accusative form is exactly the same as the nominative dictionary form. If the masculine noun were a living thing, it would get an extra ending.