shoe (n.)
Old English scoh "shoe," from Proto-Germanic *skokhaz (cognates: Old Norse skor, Danish and Swedish sko, Old Frisian skoch, Old Saxon skoh, Middle Dutch scoe, Dutch schoen, Old High German scuoh, German Schuh, Gothic skoh). No known cognates outside Germanic, unless it somehow is connected with PIE root *skeu- "cover" (cognates: second element in Latin ob-scurus).

Old plural form shoon lasted until 16c. Meaning "metal plate to protect a horse's hoof" is attested from late 14c. Distinction between shoe and boot (n.) is attested from c.1400. To stand in someone's shoes "see things from his or her point of view" is attested from 1767. Old shoe as a type of something worthless is attested from late 14c.

Shoes tied to the fender of a newlywed couple's car preserves the old custom (mentioned from 1540s) of throwing an old shoe at or after someone to wish them luck. Perhaps the association is with dirtiness, on the "muck is luck" theory.
shoe (v.)
Old English scogan "to shoe," from the root of shoe (n.). In reference to horses from c.1200. Related: Shoed; shoeing.