오늘 몇월 며칠이에요? (1) (1) is correct as you say (but 몇월 should be written as 몇 월 with a space). It is a question often asked by native speakers too because it seems to defy common sense. How it came to be 며칠 is said to be unknown - it is spelled this way primarily because it is the established pronunciation. To fully understand it, you must know some pronunciation rules, like 몇 월 being pronounced as [며둴], not [며춸]. This is because pronunciation of 받침 (consonant at the bottom) goes through "consonant folding" at word boundaries, as in 몇 월 -> [멷월] -> [며둴] (ㄷ,ㅅ,ㅊ,ㅊ,ㅌ cnanges to ㄷ; ㄱ,ㅋ to ㄱ; and ㅂ,ㅍ to ㅂ). According to this rule, if 몇 일 was the correct word, the pronunciation would be [면닐] (몇 일 -> [멷일] -> [면닐]), because the 이([i]) sound in 일 has the special effect of taking on ㄴ sound and changing the preceding ㄷ to ㄴ But the traditional actual pronunciation has been [며칠], which precludes 몇 일. It might be because of the high frequency of usage that made it change to 며칠, Whatever the reason, 며칠 is the right pronunciation and spelling, and 몇 일 should never be used. So you should just remember 몇 월 며칠 as a special exception.