"היום יום רביעי."
Translation:Today is Wednesday.
Yes, it is sometimes said. It's not common, and may sound a tiny bit funny, if you have a sensitive ear, but not more than a tiny bit.
One thing though: in spoken Hebrew I think most people pronounce the letter name דלד, no idea why. I remember being surprised, as a child, to see it written דלת.
Well, it is not Yiddish, where it is pronounced דלתּ (i.e. דּאַלעט) or דלת (i.e. דּאַלעס). One candidate (if not simply a colloquialism which wanted an additional d-sound for the d-letter) would be Italian Hebrew, where final תָּו is pronounced [d]. Surprisingly the letter is called ܕܳܠܰܕ [i.e doladh] in Syriac.
Actually, it is not all that uncommon in spoken English to use what linguists call a "left-dislocation" in certain contexts. They actually serve special linguistic functions that you might not learn about in at least some English "grammar" classes. However, since "left-dislocations" or "pre-clausal dislocations" are much more common in Hebrew (sometimes for different reasons than English) and the above Hebrew sentence doesn't even use one, it's usually more accurate to default to "Today is Wednesday".
I posted more about "left-dislocations" in the following comment thread (if you're interested): https://www.duolingo.com/comment/16827893
No, the rules are quite complicated. In the type the nouns is a noun the copula is common, usual when the subject is longish, and obligatory, if the predicate is definite too. But in this sentence הַיוֹם is an adverbial expression like מָחָר tomorrow, אֶתְמוֹל yesterday or עַכְשָׁיו now, it does not mean any more the day is a fourth day. The subjects in it is Wednesday can be expressed as זֶה(וּ) יוֹם רְבִיעִי, but if you have another element first, you can leave it out.
Well, as the subject הַיּוֹם the day is sort of short, one would rather omit the copula. If the expression around the יוֹם were longer, like כִּמְעַט כׇּל יוֹם כְּשֶׁאֲנִי יוֹצֵא בַּבֹּ֫קֶר nearly every day, when I go out in the morning, ... you would have to continue with the copula, f.e. הוּא קַר מְאֹד עַכְשָׁיו ...is very cold now, because the boundary between subject and predicate is not so clear anymore.