You can (and probably should, as some people get up in the afternoon). What you can't omit is the space between 'every' and 'day'. ;-) ('Everyday' is an adjective.)
Yes, you should include it. As you know, there are people who do not get up in the morning, like those who work on night shifts, for example.
But then you could also include "in the morning" in the question; it would be understood that your answer refers to the morning, not to any other time of the day.