First question, will they be smart enough to go into the coop at night from the lot.
It's not a question of them being smart enough, they tend to follow instinct a lot. Until they are ready to roost they tend to like to sleep in a group on the floor. That floor may be in the coop but could also be on the ground outside. I've had broody hens take their chicks to the roosts as early as 2 weeks. The earliest I've had brooder-raised chicks start to roost on their own is about 5 weeks with 10 to 12 weeks being average for most broods. Some few have gone a lot longer than that. Each brood is different. I think configuration of roosts, coop, and run makes a difference too. I've had more trouble getting chicks to go into the coop at sunset with an elevated coop than a coop on the ground.
You may get lucky but it is quite likely you will have to train them to go into the coop at night. Some people have success locking them in the coop section only for a week or so before they allow them into the run. That approach has not worked for me with my elevated grow-out coop. Mine practically always try to sleep in a group under the pop door so I have to put them into the coop each night until they all go in on their own. I've had some groups that all I had to do was put them in once and they got the message. I've had some that it took doing that for three weeks straight before the last one caught on. Usually it takes about a week.
Second, after reading some posts on predators, will they be safe inside this coop or do I need to fortify? We are in western NC and am now starting to worry about weasels sneaking in.
It's hard to say. Weasels come in different sizes and I can't tell the size of the openings in that hardware cloth or what other cracks or openings you may have. In western NC weasels are not the only risk either. You could have coyote, fox, dog, raccoon, skunk, bobcat, possum, maybe even bear. I can't tell how robust the hardware, wood, or connections are. I don't mean to unnecessarily frighten you, you will probably be OK unless you get really unlucky and get a bear but I can't honestly tell how well that coop is built.
One thing I strongly suggest is that you put an apron around it. Some critters can squeeze under the base if there is even a tiny crack, let alone dig their way in. Lay a piece of wire mesh maybe 18" wide around the outside of the run and attach it to the bottom of your fence so nothing can get through (overlap the corners). You can use wire, J-Clips, hog rings, screw with washers, or furring strips to attach it. You don't have to bury it, just lay something on top to hold it flat until grass grows up through it to hold it down, but I like to take the top 2" of turf off then put that back on top to protect it from weed eaters or lawn mowers. The idea is that a critter goes up to the fence, starts to dig, hits the wire, and does not know to back up. It is very effective and not that hard to do, especially if you try to dig straight down in rocky soil to extend a fence straight down.
