"I am going home early today."
Not unless you want to specifically talk about what happened today. For example:
今日、レストランに行った (Kyō, resutoran ni itta) — Today, I went to the restaurant.
今日はレストランに行った (Kyō wa resutoran ni itta) — As for today, I went to the restaurant.
Both are correct, except that in the second example the speaker wants to talk about how today was. It’s up to you to choose whether you want to talk about the day or rather what you did then.
Yes, 今日は家に早く帰ります is prefectly fine. You technically don't even need 家に (いえに) as 帰る (かえる) usually implies home as the destination.
Do you remember かぜがつよくふきます (the wind blows strongly) from one of the earlier lessons? The く makes it an adverb.
I believe that is because it describes an action and not an object (ie "he walked quickly" as opposed to "he is fast")
Yes, the "ku" is necessary when the word serves as an adverb, rather than an adjective
Since it's asking for " I am going home," which is present/future progressive, shouldn't you say『今日は早く帰っています』?
Person A stands up from their desk
PERSON B: "Where are you going?"
PERSON A: "I'm going home early today."
The journey has already started, it's therefore present progressive, and I think 帰っています is an acceptable translation.
"Going" in this example could easily mean either "I am going to go," which is probably the more common usage in English, or just plain "going," which is what it actually says.