If I had a run like yours I think I'd leave them out in it too. (especially in texas! Only spent part of one summer there but that was QUITE enough thank you very much!) I can see it's got good hardware cloth and the box wire looks like excellent reinforcement against dogs etc, buried wire against digging. My only question is about where the roof meets the sides, but still, I'd probably leave my hens out especially since your new dog looks like he'd at least sound an alarm if something came sniffing around, probably defend them too. (sounds like both you and he picked good people to hang out with!)
Also a fact in favor of them being relatively safe in there. If something got in during the hours it was light at all, the birds have room to evade since it looks quite large. If it got in after full dark of course, chickens just do NOT react and are literally "sitting ducks" (or hens) cause they will just sit there being eaten. Still, that cuts out a little of the dangerous time for them. Many coops and pens restrict the hen's movement to the point that they can get cornered very easily.
I'd leave mine out in that.
Also a fact in favor of them being relatively safe in there. If something got in during the hours it was light at all, the birds have room to evade since it looks quite large. If it got in after full dark of course, chickens just do NOT react and are literally "sitting ducks" (or hens) cause they will just sit there being eaten. Still, that cuts out a little of the dangerous time for them. Many coops and pens restrict the hen's movement to the point that they can get cornered very easily.
I'd leave mine out in that.