drag (v.) Look up drag at Dictionary.com
mid-15c., from Old Norse draga, or a dialectal variant of Old English dragan "to draw," both ultimately from Proto-Germanic *dragan "to draw, pull," from PIE root *dhragh- "to draw, drag on the ground" (cognates: Sanskrit dhrajati "pulls, slides in," Russian drogi "wagon;" but not considered to be directly the source of Latin trahere).

Meaning "to take a puff" (of a cigarette, etc.) is from 1914. Related: Dragged; dragging. Drag-out "violent fight" is from c.1859. To drag (one's) feet (1946, in figurative sense) supposedly is from logging, from a lazy way to use a two-man saw.
drag (n.) Look up drag at Dictionary.com
c.1300, "dragnet," perhaps from a Scandinavian source (compare Swedish dragg "grapnel") or from Old English dræge "dragnet," related to dragan "to draw" (see drag (v.)).

Sense of "annoying, boring person or thing" is 1813, perhaps from the notion of something that must be dragged as an impediment. Sense of "women's clothing worn by a man" is said to be 1870 theater slang, from the sensation of long skirts trailing on the floor (another guess is Yiddish trogn "to wear," from German tragen); drag queen is from 1941.

Drag racing (1947), is said to be from thieves' slang drag "automobile" (1935), perhaps ultimately from slang sense of "wagon, buggy" (1755), because a horse would drag it. By 1851 this was transferred to "street," as in the phrase main drag (which some propose as the source of the racing sense).
In addition to the time trials there are a number of "drag races" between two or more cars. They are run, not for record, but to satisfy the desire of most Americans to see who can get from here to there in the fastest time. ["Popular Mechanics," January 1947]