I read... on this very forum... what I consider to be an excellent idea in identifying predators and that is to scatter some fine sand or something so that whatever-it-is has to step on the sand (or whatever you use) so you can see the foot prints. Even if it is rats, they will leave foot prints. Then at least you will know.
With rat/mouse eradication, my experience is there will always be some, no matter what you do. I leave out live traps that can catch multiple animals. For mice it's a small steel box that they walk through and end up in the holding area instead of being able to go through. I just saw a nifty live rat trap in the Stromberg's catalog that is made of cage wire, they can get in but not out. For me it's just all part of having a barn that I have to check my traps every other day or so. No matter how many mice I kill, more will come.
The cement slab is a fantastic idea and one that I'll be using in my new coop. In the past though I've wired up welded wire mesh all across the bottom of the coop.