Dwarves not storing mugs in bins

Dwarves will generally sleep wherever they see fit unless they have an assigned bed, in which case they will prefer this one. Sleeping in an assigned bed does give happy thoughts, being unable to makes a dwarf unhappy.

However, you don't generally have to bother with assigning beds since dwarves will claim unassigned bedrooms for themselves automatically.

Just a side-note: married dwarves and children share beds.

There is one possible reason not mentioned above. In your dwarves' standing orders is an option disable all wood-hauling. The standing orders are where you can change settings like telling your dwarves whether to gather refuse, whether to auto-loom thread into cloth and so on.

o to get into the standing orders screen, then w to toggle wood-hauling on or off.