Learn Polish: Jutro przedstawię wszystkim nasz nowy projekt. - Tomorrow I will present our new project to everyone.

Here, wszystkim is the Dative case form of wszyscy (everyone). The verb przedstawić (to present/introduce) takes the Dative case for the person you are presenting to.

Przedstawię translates as "I will present", but there is no future helper verb like będę. Why is that?

Przedstawić is a perfective verb, meaning it describes a completed action. In Polish, when you conjugate a perfective verb using standard present-tense endings, it automatically creates the future tense. So, przedstawię directly means "I will present".

What case is the phrase nasz nowy projekt taking here?

It is in the Accusative case because it is the direct object—the thing being presented. Since projekt is a masculine inanimate noun, its Accusative form looks exactly the same as its basic Nominative form.

Does the word order matter here? Could I put wszystkim at the very end of the sentence?

You can say Jutro przedstawię nasz nowy projekt wszystkim, but Polish naturally prefers to tuck short pronouns like wszystkim right after the verb. Moving it to the very end would place a heavier, specific emphasis on the fact that you are presenting it to everyone (rather than just a few people).