Egor, I don't have an answer for first question. But you're right DATE() is available in the All list and not in the Date list. Tableau considers it a Type Conversion so you'll find it there. You can use DATE() simply by double clicking it in either list. That said I'm not certain that's what you're looking.
One other note: COUNTD() is available if you extract your data (MS Jet engine doesn't support that function, Tableau's data engine does).
Many thanks, Shawn, for reply. COUNTD(DATE([date_modified])) works exactly as needed.