distances - How near to Earth do comets pass?

The answer is that they don't come very close. As Wikipedia notes,

It is rare for a comet to pass within 0.1 AU (15,000,000 km; 9,300,000 mi) of Earth.

Even better, though, is an actual list of some of the closest approaches of comets to Earth. The closest one listed, Comet Lexell in 1770, came 0.0151 AU away from Earth. The list only shows 20 comets have come with 0.1 AU of Earth. So Earth has not had a close encounter with a comet (like Mars is having) in a long time, if ever.

This site says that C/1491 came within an incredible 0.0094 AU in 1491, but I'll take the IAU site over it. And it does admit that that comet has an uncertain orbit, so that figure could be wrong.

But when comets come even a tiny bit near to Earth, there are some articles written about them. See here and here.

As for the distances to famous comets:

Closest Ever

• Halley's - 0.0334 AU


• Great Comet of 1760 - 0.0682 AU


• Tempel-Tuttle - 0.0229 AU

Also, some near-Earth asteroids used to be comets. Extinct comets have lost a lot (if not all) of their ice and so may be classified as asteroids. This blurs the lines between the two groups. Some extinct comets have become near-Earth asteroids. This explores the issue pretty well. So if you include extinct comets as "comets," you may have comets that have come even closer to Earth.