Stamina runs out from the point you get encumbered to the point where it hits 0. That's why you can walk a little before your character is too exhausted to move further.

For stamina to regain, you need to reduce the weight you are carrying. If you managed to have a chest in reach, you can put stuff there; otherwise you have to throw it on the ground. If it's within the radius of a workbench, it won't despawn at all, IIRC. If it's in the wild, it should despawn after a couple of in-game days.