1) 전화번호 means "phone number" and you're not being specific if it's landline, cell (mobile), etc. 핸드폰 is "cell phone (mobile phone)" and it's the transliteration of "hand phone" in Korean version of English. Many Koreans also say 휴대폰 for a "cell phone". I would say both are used widely. Koreans abroad (especially those in the US like I am) say 셀폰 (transliteration of "cell phone"). 전화번호가 몇 번이에요? = What's your phone number? 핸드폰 번호가 몇 번이에요? = What's your cell phone number? *Instead of the '몇 번이에요?' ending, you could also say '____ 가 어떻게 되요?'. 2) (오늘이) 며칠이에요? and (오늘이) 몇 월 며칠이에요 pretty much mean the same thing, you will hear both but to be more specific, the first is just asking what the *date* is (just the date without the month) and the latter is asking what month + what day it is today. Now, if you want to ask "what day is today? (what day of the week)", you can say 오늘(이) 무슨 요일이에요?

1) nothing is common. Koreans do "(전화)번호가 어떻게 돼?(what is your phone number)" "(전화)번호 좀 알려줘(tell(teach,show) me your phone number)"