They are both technically correct to me, but the first sentence is in subjunctive mood (ie. your opinion) and the second is in indicative mood (a fact). The unfamiliar grammar of subjunctive mood in English makes many people mistake it as "old", "overly-formal" or "poetic".

Both are correct, they have the opposite meaning to each other. "The room need not be cleaned" sounds very formal or old fashioned to me, but it is perfectly OK to use. It just might sound a little bit too formal in the wrong setting. "The room does not need to be cleaned" is another option.

Yes they are both correct. But the second sentence is more appropriate if written like this 2.The room needs cleaning

yes.but if u use it in this form,it would be better The room doesn`t need cleaning or the room doesn`t need getting cleaned.