late 14c., "overgrown with trees and shrubs," from wood (n.) + -y (2). Of plants, "having a stem of wood," from 1570s. As a name for a kind of station wagon with wood panels, by 1961, U.S. surfer slang (real wood exterior panels were rare after 1951 and the last use of real wood was in the 1953 Buick Roadmaster Estate Wagon). Slang meaning "erection" attested by 1990 (for hardness).
To rehearse; practice one's part, role, etc, esp to do so alone and rigorously: Bix did plenty of woodshedding, playing alone/ I just learned a new technique and I've got to shed on it
[1930s+ Jazz musicians; fr the woodshed as the traditional place where one could be alone to work, think, smoke, etc]