a) near 2000. It may have been 1999, it may have been 2001. Maybe even 1998. Uusually, it's used for something further back in time when you'd be less certain of the exact time. 2000 was pretty recent and it shouldn't be so hard to figure out if it was 2000 or 1999 or 2001.
b) You can say "any time of the year" to mean at any point in time. I'm not familiar with "anytime around the year" as a phrase.
c) "around the clock" means all the time - 24 hours a day. Or, for 24 straight hours - they danced around the clock. I'm also not familiar with "anytime around the clock" as a phrase.