Native English speaker (Canadian) here..."what is the day today?" sounds like you're asking which day of the week it is (Monday, Tuesday, Wednesday etc). not what the date is (for example: May 4th, December 25th etc).
No. "Detumett" is a misspelling, it would be "datumet". I also can't give a hard grammatical reason why, but "Vad är datumet idag" doesn't sound entirely natural to me as a native speaker, or at least not as the most common way you would say it.