I'm glad that here the translations "It's finally the weekend!" and "Finally it's the weekend" are accepted (as opposed to a sentence in the last lesson). However, "Finally it is weekend!" should not be accepted. The weekend is never referred to indefinitely in English unless it's plural. In the singular it's always definite. Thus, "Finally it is weekend!" just seems strange.
How about "At last, it is the weekend!"? Isn't "végre" something like "at last" or "at the end"?
I have only heard "végre" used to mean "finally". In English, such concepts are very interchangeable, but I don't know if Hungarian sees a more hard tie between terms like "végre", "végül", etc. and their English counterparts. It may be that they are similarly interchangeable. Hopefully one of our native Hungarian-speakers will pipe in and help clarify.
"Finally it is weekend!" needs an article before the singular "weekend."
Furthermore, someone saying "Finally X" isn't going to follow that word with "it is". If they're exhaustedly relieved that something has finally come, they're going to say it quickly instead of slowly by using "it's" instead.
You're absolutely correct on the article. As for "it's" vs. "it is", yes, "it's" is going to be far more common, but "it is" is entirely valid, and does get used.