Yes, mice and rats are smart and will learn to come at dusk or later. That's what happened in my run/coop. Cats and chickens won't get every mouse anyway. Traps can get messy and could be problematic with your birds.
1/2" or 1/4" hardware cloth is your best bet. I found the weak spots (areas I was too lazy to cover with it when it was built) and covered them up- problem solved from here on out. Don't forget to include the ground underneath the run and coop.