I think Trinoc is seeing nose hairs. They certainly look similar.
No, I've heard this stated at least as far back as my biology teacher at school. You know what it feels like to have a single hair, or even a bit of dust, stuck between the eyeball and the socket ... imagine what it would be like if every eyelash you'd ever lost in that way was still in there.
Hair is one of the most durable parts of the body. After teeth and bones it is the part most likely to be preserved when bodies are dug up after hundreds of years. If you eat anything with hair on it, the hair is not digested in the acid and enzymes of the stomach ... look at the way cats vomit up furballs rather than digesting them. You can just about get hair to dissolve if you boil it in caustic soda. The much more benign tear fluid around the eye would have no effect on hair material (keratin) at all.
Eyelashes are eliminated from the body, and unless they find their way out from the eye itself this is via the tear duct at the nose end of the eyelids and into the cavity behind the nose. From there it can either exit out of the nose, into the mouth or down to the stomach and excreted with other indigestible fibre.