The BLS publishes a table of internationally comparable unemployment rates, which take into account differences across countries in definitions and concepts. So does the OECD.
The “official” unemployment rate doesn’t take into account whether or not you are receiving or have exhausted unemployment benefits. A person’s classification is based on a series of questions about his or her “major activity” during the survey reference week.
Sort of. The "series of questions" are asked on the UI benefit forms.
Once you've exhausted your benefits, you cease filling out the forms...