Correct. Perfect English writing. If you want informal, then "I was sick many times..." is good enough. "Got sick" is also acceptable in speech but is bad English spoken by children etc. See, http://www.tudou.com/playlist/p/l2350485.html to understand what I mean. I don't want to sound hoity doity here but we must speak good English at all times.

you could say that. i guess. but if you want to sound more native I (personally) would say something like: I got sick a lot this month because of the bad weather. or to shorten it a bit you could say "I got sick a lot this month" saying "fell ill" is almost never said unless you want to act old and stuffy, hahah. You would hear that in a novel, book, or play that was set decades ago. you can say "ill" if you want, but again, its uncommon.

A native speaker would say "I got sick many times this month". I fell ill is a bit too formal.

I would say: "I was sick all month because of the cold weather."

Yes, it's correct. You should wear warmer clothes.