Translation:The sky will be clear tomorrow.
Yes and no. It comes from the verb 晴れる（はれる）, "to clear up", and a clear sky is more or less the same as sunny weather. After all, without any clouds, all you'd normally see is the sun. The kanji itself is even made up of 日(sun)＋青(blue)! はれ in itself though can also be used more figuratively, such as something that's public, or a clearing of suspicion/misunderstanding.
It is possible. I think "The sky is clear tomorrow" should be accepted and "The sky will be clear tomorrow" as well.
I would say that there is a slightly different nuance when using "is" vs. "will be". For me "is" implies more certainty about the result, whereas, "will be" is less certain than "is" but still pretty certain. But this is a pretty subtle distinction that won't make much of a difference in everyday speech.
As another example of the simple present vs using 'will', "They meet next year" vs "They will meet next year". Omitting the "will" seems to indicate more emphasis on the certainty of the meeting next year, at least to me.
You can see "languagelog.ldc.upenn.edu/nll/?p=897" for a linguist's point of view.
Probably because that implies that it is not currently clear right now . Adding "be" before "clear" removes that implication.
As a somewhat similar example: The rain will stop tomorrow VS There will be no rain tomorrow. The first version sounds as if it is currently raining, but the other just lets us know that it won't rain tomorrow
No idea. As pointed out in earlier comments; そらがはれます literally means "the sky will be clear". However, "it will be sunny" should be fine, as that comes down to the same thing.* You do not (ever) need to say "the sky will be sunny", that's indeed pointless.
*caveat: at least in Japan and most other places on Earth. If you live above the pole circle, a clear sky might not necessarily mean there will be sun.