hi, Good evening lemon tea. I feel weird when someone speaking to me 学校何日开始放暑假？I think natural way to say that 学校哪一天开始放暑假？学校几号开始放暑假？ 何日 is very formal writing that appears on classic and titles. it is shorten phrase that indicates which date. but in google translation there is connotation that means day too. but in modern chinese. 日 i think it means date. I forgot the second question now I add on. 何日 and 何时 difference is what date and what time of the day. 时 is short for 时候 ，时间。时候 is about morning, afternoon, evening. and 时间just any time.

何日 is the same as' when'. But generally speaking, The more colloquial expression is 什么时候，

1. It work for both. You could answer May 25th or Friday. 2. Commonly speaking, 何日 means what date, and 何时 means what time. But 何时 sometime means what date as well. 何日 never means what time. 3. 何日 and 何时 pretty classical. We don't use it so often. Mostly in some idiom, like 曾几何时 (once upon a time).

何日 is archaic. I would expect someone from Ancient times to ask date with 何日。And my brain might switch to the lunar calendar. There was a song with this word though, which was really popular those days.《何日君再来》When Will You Come Again

何时：是哪一天，什么时候。 何时：什么时候（表疑问，时间难以确定。）