Hello! I agree, your coop is sure pretty! About snakes. Some climb better than others, garters aren't the best climbers but they certainly can and do if they want to. I have one in captivity here at work. I would say they can't climb straight up 1/4" hardware cloth but if it is slanted, wavy, or a little imperfect, they will certainly try. Also, garter snakes aren't known for eating chicken eggs, so may be after worms, insects, frogs, and small mice. They are great hunters...also, chickens will sometimes eat small snakes so good luck to the babies that wander in! Garter snakes give birth to live young so the mom has to give birth nearby for there to be a lot of them, and around here, late summer and fall is baby snake season.
That said, many other snakes are most excellent climbers, as you stated. Especially black rat snakes and black kingsnakes. Those two are well known for climbing and eating baby birds, eggs and also chicken eggs if they can get them. I would definitely try to keep the big ones out. These "black" ones can climb nearly any rough surface so minimizing gaps and holes will go a long way in keeping them out. Also, keeping mice out will help. So having them "around" is good! " IN " the coop, is not so good. As for rattlesnakes, you're on your own there... not the best climbers I think, but I'm not a rattler expert! Best of luck to you!