I think an important part of the question that isn't answered in the FAQ concerns confirming that we haven't lost count at any time in the last 1500 or so years we've been using the AD calendar.

The first way we can do this is by examining archived records. Certain institutions, notably the Catholic Church, have been maintaining a solid chain of records basically since they were founded. If you felt like it, for instance, you could climb down in the Vatican archives and count to confirm indeed that since the year 600, there have been 1,415 Easter celebrations, or whatever.

We can bolster this by astronomical observations. If we know Haley's comet appears every 83 years, and we have contemporary accounts of its observation in the years 560, 643, 727, etc. all the way through now, we know that we haven't missed a year or accidentally counted one twice.

Well technically there was no year 0, the year system goes from 1 BCE to the following year 1 CE. By definition of the calendar we haven't missed any years.

Also most scholars agree that Jesus wasn't born 1 CE, as you might think, but generally his birth is thought to be a few years before. So you might look at that as missing a few years, but it's not like (for example) year 273 AD/CE never existed... They just dated their calendars using different means at that time.

For further reading see the responses in the FAQ

We lost 10 days in the 1580s. Check out Gregorian Calender.