I was always told that if a single string falls off your Tzitzis you should put it in a Sefer so that it will end up in Shamis. My question is: Does anyone know a source for this?
The Mishnah Berurah writes that the Maharil recommended using old tzitzis as a bookmark or for another mitzvah, because we have a rule in the Gemara that one should try to take an object used for a mitzvah and use it for another mitzvah ("ho'il v'isavid b'hu mitzva chado, yisavid b'hu achariso").
Your second question is addressed in the Shulchan Aruch, O.C. 21, 1-2. The begged can be thrown in the trash. Regarding the tzitzit, three opinions are presented: you can toss them in the trash, you should dispose of them in a more respectful manner (e.g. toss them in the bushes) or you should put them in shaimmes.