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Thank you, Rusty, for your detailed answer. I would be grateful if you could answer a couple of follow-up questions.
From the point of grammar,
1. The answer to "¿qué día es hoy?" would be, for example, "Hoy es dia 15" or "Hoy es lunes". "Hoy" can be omitted in which case the replies are "Es dia 15" or "Es lunes" or just "Dia 15" or "Lunes". Is "hoy" (adverb) the subject of the question?
By analogy, in the English phrase "Today is Monday" “today” is a subject as well. Is it possible to retain the sentence structure as in "It's Monday today" in Spanish? All I can think of is placing "hoy" at the end: "es lunes hoy".

It seems to me that the concept of itness is somehow contained in the verb, it is expressed by the verb (given a context). Am I wrong?

2. It seems to me that the answer to ¿a qué día estamos hoy? would be (Nosotros/nosotras) "estamos a lunes hoy". The subject is either "nosotros" or "nosotras" (in case of a group with mixed genders the nosotros prevails). Is it the standard way to answer this question?
Please correct me if I am wrong in my assumptions.