"Today is a beautiful day."
Translation:He lā nani kēia lā.
Today is actually just "kēia lā". The "i", when used with "kēia lā" marks it as a time stamp - the time frame when something happens. In this case, you are not saying that today is when the beautiful is happening (though that is true). In this case you are specifically saying that today is the thing that is beautiful.
In English we don't mark time stamps differently from subjects so "today" doesn't change between "today is beautiful" vs. "today it is beautiful". But in Hawaiian, you have to add the "i" marker when using it as a time stamp instead of the subject. Since we don't have to do that in English, we have to put a little extra attention to whether "today" is being used as a thing or as a time stamp when saying it in Hawaiian.
Yes, that's correct with one small change that the word for word translation is "A day beautiful this day."
Though I could see arguing that when "He" (A) begins a sentence it indicates that the sentence will be an equivalence and thus in this specific situation could be translated as "Is a": "Is a day beautiful this day."
It's a word or phrase you add to a sentence to give a time frame to what you are saying, like "today", "5 minutes ago", "when I grow up." If I say, "Today was beautiful," my sentence is actually about today. If I say "Today it was beautiful", my sentence is about "it" and "today" is just the time frame. Part of the confusion comes that in English we often use that "it" to refer to "today" and it's sort of playing a dual role in that sentence. But grammatically, one version uses "today" as the subject and one uses "today" as a time stamp. The same grammatical difference can be made in Hawaiian, so you have to watch to see if it is in the position of the subject or if it is marked (with "i") as a time stamp.