Learn Farsi: مسیر این رژه به طرف خیابان بزرگ نیست. - The route of this parade is not towards the big street.

It uses an Ezafe! The phrase be taraf literally means to the side or to the direction. To link it to the destination, you must add the Ezafe (a short e sound), making it be taraf-e khiābān (towards the street).

Since the current lesson covers be su-ye, be samt-e, and be taraf-e, could I use any of them here?

Yes, they are essentially interchangeable. You could easily say be samt-e khiābān-e bozorg or be su-ye khiābān-e bozorg and the meaning (towards the big street) would remain exactly the same.

Where exactly do the hidden Ezafe vowels go in this sentence?

There are three Ezafe links here: masir-e in rezhe (the route of this parade), be taraf-e khiābān (towards the street), and khiābān-e bozorg (the big street). Notice that the words in (this) and rezhe (parade) do not take an Ezafe after them.

Why does the verb نیست (nist) come at the very end of the sentence?

Farsi is a Subject-Object-Verb language, meaning the main verb almost always anchors the end of the sentence. Unlike English which puts the verb early on, Farsi literally says: The route of this parade towards the big street is not.