Foul smell in my bathroom when I run water anywhere in my house
First I'd want to confirm that it's really from the waste system (as I would think).  When there's no smell, plug another sink and run water to put some in the basin.  Stop and verify still no smell.  Then drain the basin and see if the smell arises.  If that happens, you can be sure it's related to the waste system.

Can you plug the sink temporarily (both the drain and the overflow)?  If you do so, does that prevent the smell?  It would be nice to isolate it to a single location.  Is there a tub/shower?  Could it be coming from there?

If it's from the sink, I'd think there's either improper venting or incorrect p-trap in the bathroom.  Can you show a picture of the pipes under the sink?  If it's not set up correctly, then water running down the pipes from elsewhere can siphon the water from the p-trap, allowing sewer gases to escape the drain.