For the first, you can also use "What is today's date?"
Your second is about as good as it gets.
Hi, I teach English in Japan.
Here is my questions;
1. What kind of expressions/questions do you often use when you want answers like "April first"? I could think of "What is the date today?" or "What day is it?" Is there any other expressions you could use in classroom?
2. What kind of expressions/questions do you use when you want "Wedensday"? I believe "What day of the week is it?" is good, but is there any other expressions?
Thank you in adavance.
What day is it today?
Please allow me to ask a question here.
Is "What date is today?" natural English?
What is today's date?
What is the date today?
What date is it today?
"What date is today?" is unnatural for me.
As far as different ways of asking for a day of the week, it depends on the level of your students. You didn't say you were necessarily asking only for the name of the day today, so you can elicit other days of the week in many different ways:
What day will it be the day after tomorrow?
What day was it yesterday?
What day was it the day before yesterday?
To elicit dates, you can ask all kinds of questions:
What date is your birthday?
What date is your parents' wedding anniversary?
What date was it last Tuesday?
What date will it be on Friday?