Actually you can. はれ as a noun here. Compare the noun form and the verb form:

• 今日は晴れです
• 今日はくもりです

• 今日は雨です

• 今日は晴れています

• 今日はくもっています
• 今日は雨が降っています

Yeah

I think you mixed up the two concepts. …が…に…いる is the construct for the existance of animated subjects.

However …ている has two different meanings and is different from above.

1. Action verb + ている means the continuous action of the verb..e.g. 見ている "is looking," 食べている "is eating."

2. State verb + ている means the continuous state of the verb. e.g. 始まっている=始まった "something started/is starting," 着いている=着いた "someone arrived," 終わっている=終わった "something finished."

います does not only mean the existance of animate subjects, but it is used as a auxiliary verb in ている, which denotes either the main verb is in a continuous state, or is in a continuous action. Please see my other comment for details.

も(particle): also. もう(adverb): already.

Completely two different things.

Yes. も modifies きょう so も needs to be written at the end of きょう.