Our verb "to drag" is rich in metaphor: dragging something; drag racing; drag queens; take a drag. In English, both verbs "to drag" and "to draw" are closely related. In German, the related verb tragen is more limited: it means to wear, to carry, to bear, and has other supporting roles; they have another word for the basic idea of dragging, ziehen, which means to pull, to drag, to draw, to haul, etc.
Here's the plausible origin of the term "drag race:"
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). linkThere is an old related word "dray" which meant a wagon for hauling stuff; a "drayman" was a "truck driver."
The origin of the term "in drag" is controversial:
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. linkThe "dragging on ground" could be related to women's costumes being longer and having trains. The Yiddish origin sounds better, also given that the German word for costume is Tracht which in turn relates back to tragen.
 drag (v.)
The root of the verb "draw" comes full circle back to "drag:"
 The German verb "ziehen" is murkier. My unlinkable Duden 7 says:
ziehen: Das altgerm. Verb mhd. ziehen, got. tiuhan, aengl. teon (vgl. aisl. togenn "gezogen") gehoert zu einer idg. Wurzel deuk-ziehen; vgl. aus anderen idg. Sprachen z.B. lat. ducere "ziehen"translation:
Pull: The old Germanic verb "pull"; gothic tiuhan, old English teon (cf. old Icelandic togenn "pulled") belongs to an Indo-European root "deuk-ziehen." See, in other Indo-European languages, e.g. Latin ducere to pull, to lead.
I think there's something still missing.