Translation:I am going home early today.
Even if we had kanji, 早い is both fast and quick, no? (which then becomes 早くwhen used with a verb)
早 is more associated with being early, and 速 is more associated with speed. Even though you could translate "早いですね" as "you are fast", it still holds a heavier nuance of "you were early" (or earlier than the speaker expected)
Im coming/going home early could both be valid answers, depending on the listener's point of reference.
Why isn't the the "-te imasu" form of the verb used here? "Going" is a progressive verb form, right?
"I am going" is future tense here, a shortened version of "I am going to go".
No because as you've pointed out hayaku is an adverb and is describing/modifying the verb. The particle を follows the direct object of the verb - an adverb cannot be the direct object of a verb - it's a describing word.
Didn't know that はやくnot only means hurry but also early in this context. Nice.
If you wanted to spesify "early" rather than "quickly", could you say 早くに, or is the distinction between 早く as an adverb and adjective meaningless in japanese?
The distinction between adverb and adjective is most definitely not meaningless, but you seem to be confused about the difference between い-adjectives and な-adjectives.
The two types of adjectives are very different from each other, with な-adjectives acting more like nouns (they function as nominals and conjugate as nouns do), and い-adjectives acting more like verbs (they function as predicates and have their own conjugations).
い-adjectives are characterised by the い sound that they all end with: 長い（ながい、long）・短い（みじかい、short）・広い（ひろい、wide）・早い（はやい、early）・速い（はやい、fast）. Unlike な-adjectives, these can modify nouns directly, e.g. 長い本（ながいほん、a long book）. Also, they cannot be used with the casual だ copula (though they will still be used with です if speaking politely). They conjugate as follows:
Most な-adjectives on the other hand, do not end in い, although there are a few such as きれい （pretty, clean）and 嫌い（きらい、disliked）that do. Examples include: 静か（しずか、quiet）・有名（ゆうめい、famous）・大丈夫（だいじょうぶ、safe, alright, okay）・大切（たいせつ、important）・簡単（かんたん、easy）. In order for these to modify nouns, you need to append a な to them, e.g. 静かな町（しずかなまち、a quiet town）. Conjugations are as follows:
早くに can never be used because 早く is not a な-adjective, it is the adverbial form of the い-adjective 早い. If you wanted to specify early or quickly (if it's not obvious from context), you would simply write it with the kanji 早くor 速く respectively.
"I am going home early today." Since the answer has "going," why wasn't the te form of "kaerimasu" used? Actually, shouldn't also be "coming" instead?
Since hayaku is an adverb, shouldn't it be beside the verb kaerimasu?
Your translation is fine, but adverbs don't have a specific "place" in a Japanese sentence.
since the system of particles make sentence ordering flexible, adverbs can be placed anywhere in the clause that it applies to as long as it comes before the verb that it refers to.
"Go home" is a very natural and common translation of 帰ります (kaerimasu), though I think if you submitted an error report for "I am returning home early today" it would be added to the database if it wasn't accepted.
- to return; to come home; to go home; to go back
I think you can go home early without going home soon, which is why it's not accepted. For example, if you work 9-5 and you come into work at 9 and say "I'm going home early today at 3", the time you are going home is still hours away.
Yes Isola, you're right, they don't mean the same thing. Soon means in the near future or even imminently, early means before the time that you usually do something or are expected to do something - early is more like a change in schedule.