The calculator counts the whole calendar days between your birth date and the chosen date (today by default). Weeks are days divided by seven; months are full calendar months. Hours, minutes and seconds are simple multiples of the day count, so they are exact only to the day. The milestone is the next round number of days, such as 10,000 or 20,000, that you have not reached yet.
About 10,957 days, depending on how many leap days fall in the period. Turning 10,000 days old happens at roughly 27 years and 4 months.
No. On the day you were born you are 0 days old, and the next day you are 1 day old.
See also: Age Calculator · Days Between Dates · Add Days to a Date