Yes, they were used, although probably a bit more often with horse archers who needed to have hands free for reins and the like. I’m pretty sure they evolved from an earlier practice of storing the bow in the quiver, as we can see here in the upper-right:
You’re right about the stringing and unstringing, since keeping a bow strung all the time is bad for the string and thus the power of the bow. However, if you were expecting a battle or an ambush or the like, you don’t want to be in a position of having to stop and string your bow, so you’d string your bow in advance and then you’d need a way to carry it around, hence a sheathe.
Carrying them on the shoulder is a really bad idea, because moving around with them that way is very awkward and it’s super-easy to catch the bow or the string on something or someone around you.
Longbows are a bit unusual, because they were long enough to be carried like staves, albeit with a cloth cover over them. But you couldn’t really do that with other kinds of bows.