... just what is that grey or brown stuff in the crazing lines of my 1920's
ceramic (full earthenware - not glazed iron) bathroom fixtures.  Does the
darker stuff signal an internal chemical change (e.g. exposure to air or
moisture) or is it waterborne deposit from 'washing my hands in dirty water'?

And, has anyone figured out how to remove it?

They're dirty cracks, and you're stuck with them.

My bathroom sink (originally from E 90th Street or somewhere up there) has
them too.

Try to think of them as patina, or some other locution which covers
cigarette-burned chair arms, dented car doors, liver spots, varicose veins,
crows feet, gray hair (wait a minute: yours has always been Gray hair, hasn't
it?), extended forehead,  etc.

Or you could spend $3000 on a nice new pseudo-Deco porcelain lav, plus
several hundred more for nickel-plated reproduction semi-antique faucets that
aren't quite right and have the wrong level of gloss on them. But they don't
leak.

Ralph