Do these stay in your ears when when running?

Yes, it is good for running. It is not noise blocking, I still can hear surrouding sounds like traffic so it is safe for street run.

Can you wear these in the shower?

The "waterproof" is misleading, as it is in reference to the mic. However you would still need to worry about the connection point with the device you are plugging into.After using them for a while my opinion would be they are intended to merely be resistant to sweat and a wet cloth used to clean it, but I would never go beyond that.Regardless however just remember that water and electricity really never mix well. Good luck and hope this answer helps!*EDIT* After looking over the ad again I have realized that it says waterproof on the ad, but not on the actual box. This is misleading.The description is as follows:IP57 Water ResistantFitted with specially designed water-repellent acoustic fabric, the STH30 is designed to perform in any conditions. Great for exercising. Great for anything.