Shower head is the physical object/plumbing part that you can touch.

shower (verb) - to wash yourself by using a shower shower (noun) - a device that produces a spray of water for you to stand under and wash your body shower head - a device attached to the plumbing that causes the water to spray

The showerhead is the plumbing piece that causes the water to spray nicely. You only say "-head" if you are talking about that object.

This depends on the context. "I took a shower" or "the shower head was blocked so the water trickled out"