I can't think of any off the top of my head, but might reply again later. You could also check out some 四字熟語, although those are more like proverbs rather than words. Eg. http://www.edrdg.org/projects/yojijukugo.html
I would translate のこと as ‘regarding’ or ‘about’. It shouldn’t really change the meaning but is just a slightly more indirect way of saying ‘the weather will be x’. Maybe you’ve also heard of あなたのことが好き(すき)です as a way to confess your feelings. The meaning should be the same as あなたが好きです but it might be perceived as more indirect and polite I guess.
While there seems to be a more specific explanation in this case, I believe it’s fairly common that for onomatopoeic or mimetic words with repeating syllables the last one can be replaced by り、ん or something else. E.g. ぺたぺた、ぺたり、ぺたん、ぺたんこ. So your result is close enough I’d say.