The origins of "bed" have more to do with gardening than with sleeping. Bed comes from the Teutonic word "bhedh" (to dig). Historians explain that, at one time, resting places for both animals and people were dug out of the ground. These resting places were the models for the flower beds of today.
"Garden bed" has been used by the English since at least AD 1000. And "to bed out" plants was a common term in garden manuals 600 years later. Shakespeare's "bed of roses" is figurative for the comfort you'd imagine roses would get by being tucked in.
The word "litter" is related to the French "lit," for bed and the English verb "to lie." In the 1300s, a litter was a curtained stretcher that carried nobles.
For others, a bed was a pile of hay. About 100 years later, the sense of litter was extended to mean "bedding material." As the bed was literally thrown together, though, a more pejorative sense developed. By the 18th century, "litter" meant any disordered array of things. Eventually, the word's meaning was transferred from what was scattered about to that which lies upon it: newborn animals. This last sense of the word is a brood of young.
SOURCES: 'Dictionary of Word Origins,' by Joseph Shipley; 'The Barnhart Dictionary of Etymology,' by Robert Barnhart; 'The Encyclopedia of Word and Phrase Origins,' by Robert Hendrickson; 'Webster's Word Histories.'