"It is sunny today."
Those are both correct, but that's 晴れて＋いる, which differs in meaning from using 晴れて on its own (adverbially). In hindsight I should have added that to my previous comment, so thanks for pointing those out.
A conjugation of 晴れる can indeed be used as well, though you'd put that in past tense (even when describing a current situation): e.g. 晴れた空 = a clear sky. That's because this verb describes a process, rather than a state of being. It's comparable to being hungry, which is おなかがすいている (-te form +iru) or おなかがすいた (past tense, yet translated as "I am hungry").
You're right that that's how we talk and that it would be understand in a normal conversation. A computer is checking your answer, though, and how can the computer know that you understood the word 今日 (kyou) and just decided to leave it out because it sounded more natural rather than that you left it out because you didn't understand it? It can feel awkward to use English that seems unnatural to us, but we're not being professional translators here, we're trying to show that we understood the Japanese perfectly. If you were translating something professionally, your way would be the better way.
In this sentence, 晴れ (hare) is a noun meaning "sunny", and です (desu) is the copula meaning "is", so "it is sunny".
If you use the verb 晴れます (haremasu), you are talking about the future (it will be sunny). You need to use the present progressive form to say that it is currently sunny: 晴れています (harete imasu) = It is sunny.
Thanks for that! Though I am reading here (https://www.coscom.co.jp/japaneseverb/japaneseverb01-jpr.html) that ます is not necessarily future tense but rather non-past. Is 晴れます then just less likely to be interpreted (as us translators would like) as “it is sunny” rather than “it will be sunny”? And that, to be precise, we would use the present progressive form て います?
I don't know how to explain it very clearly, but if you want to use the -masu form in the present tense, you're using it to describe a habitual action.
りんごを食べます。 (ringo o tabemasu)
[future] I will eat an apple.
[present] I eat apples (as a habit).
If you want to say that you are eating an apple right now, you have to use the present progressive.
りんごを食べています。 （ringo o tabete imasu)
[present progressive] I am eating an apple (right now).
It's the same with 晴れます. It's not a habit, it means that it will be sunny in the future. If the sky is currently in a state of being sunny, you need to use 晴れています。
Weblio has a lot of examples:
It is clear right now.
Will it be clear skies tomorrow?
THANK YOU SO MUCH! What you're saying fits with what Duo accepts and with what I've seen in my language journey so far. ます is both a statement of the future and a statement of a habitual action in the present. As you've pointed out, 晴れます cannot be interpreted as a habit so it would mean unambiguously that "it will be sunny"!