I see that nobody yet have tried to correct your text, so here are some comments as a trial (as a foreign learner!!!)

나랑 놀이 줘요
> seems incorrect. Literal meaning would be "Give a play with me". If u want to say "play with me", it would be "na.rang nolla-jweo" (cant type in Korean right now, sorry...). As panmal, of course.

안녕하세요^^
> seems correct
오늘은 날씨가 방콕에서 너무 시원한
날씨가 방콕에서 너무 시원해
> 시원한 would be used before a noun + i am not sure whether 시원해 can be used when referring to the weather
오랜만에 이렇게
> u got an adverb but no verb...
그러나 난 지금 심심해
> seems very correct to me
여러분들 누가 나랑 놀이 해줄래? ㅋㅋ
> I think 들 is unnecessary if not weird with 여러분, because 여러분 means "everybody, you all" + same remark on the verb in the sense "play with me" + 여러분 is formal and may not get along well with panmal style
정정해 도와주세요 ?? 감사합니다 :)
> in the sense "correct my mistakes", i think it shud be "kochyeo juseyyo"

Help needed from a native speaker! ;-))