Learn Icelandic: Hver er dagsetningin í dag?

In Icelandic, when you ask "what is [noun]", the question word usually has to match the gender of the noun. Because dagsetning (date) is a feminine word, we use the feminine form hver instead of the neuter form hvað. Think of it as asking "which is the date?".

What does the word dagsetningin literally mean?

It breaks down into dag (day) and setningin (the setting). So it literally means "the setting of the day". Notice the -in at the end, which is the attached definite article meaning "the".

How does í dag relate to the rules for seasons and years?

The phrase í dag is a fixed temporal expression for "today". Just like we use í for the current day, Icelandic also uses í plus the accusative case for the current season or year, such as í sumar (this summer) or í ár (this year).

If I want to answer this question with "It is...", which pronoun do I use?

You would use the feminine pronoun hún (she/it) and say Hún er... because the noun you are replacing, dagsetning, is feminine. You could also just start your answer with Í dag er... (Today is...).