It means just what it says. In the common calendar, the year after 1 BC (or BCE in the new notation) was 1 AD (or CE in the new notation). There was no "January 1, 0000".
No, it isn't, since you explicitly said to start the first century on the date that doesn't exist. What does that even mean?
The point is that some days got skipped over the centuries, but there's no need to make the Centuries have weird boundaries.
That's not what the poster I originally responded to is saying. He's saying the 1st Century should start on a nonexistent day.
That allows for consistent zero-indexed centuries. It doesn't have any other practical consequences that matter.
10 C = 50 F = 283.15 K
1 = 0.999…
Things can have more than one name. The existence of the year 0 CE is not in question. What’s in question is whether that’s a good name for it or not.