"building for storage," 1855, earlier "light, temporary shelter" (late 15c., shadde), possibly a dialectal variant of a specialized use of shade (n.). Originally of the barest sort of shelter. Or from or influenced in sense development by Middle English schudde (shud) "a shed, hut."
"cast off," Old English sceadan, scadan "to divide, separate, part company; discriminate, decide; scatter abroad, cast about," strong verb (past tense scead, past participle sceadan), from Proto-Germanic *skaithan (source also of Old Saxon skethan, Old Frisian sketha, Middle Dutch sceiden, Dutch scheiden, Old High German sceidan, German scheiden "part, separate, distinguish," Gothic skaidan "separate"), from an extended form of PIE root *skei- "to cut, split."
In reference to animals, "to lose hair, feathers, etc." recorded from c. 1500; of trees losing leaves from 1590s; of clothes, 1858. This verb was used in Old English to gloss Late Latin words in the sense "to discriminate, to decide" that literally mean "to divide, separate" (compare discern). Hence also scead (n.) "separation, distinction; discretion, understanding, reason;" sceadwisnes "discrimination, discretion." Related: Shedding. A shedding-tooth (1799) was a milk-tooth or baby-tooth.