A chronic nail-biter will bite their nails down so low that it leads to bleeding around the nails (bleedy-looking).

It's more of Holden's unsophisticated jargon. He's saying that her fingers are damaged from chewing and that they look like they've been recently bleeding. By the way, while I don't have a problem with you asking this many questions about the book, I would suggest closing previous questions by selecting Best Answers. It allows people to search for open questions more easily (and it also helps me, of course!). I'd also recommend not worrying too much about specific words in "The Catcher in the Rye". Holden Caulfield, you've probably noticed by now, makes up a lot of words or uses words in nonstandard ways. They're supposed to simply be figured out in context even by native speakers, and since your English appears to be very good, I'm assuming you already had pretty good ideas of what these words meant. Of course, if you are completely stumped, then like I said, keep asking questions. But don't second-guess yourself too much!