Your fencing sounds good, 6 foot is pretty high, 5 is not bad either and since you'll be watching them as much as possible, that will help.
I'm not sure if a full grown cat will get a full grown chicken. Maybe search the "Predator and Pests" forum on "cats" and see what pops up. That's what's great about BYC is nearly anything you need to learn, you can find some thread that's discussed it at least to some degree. Every one of us was new once so no worries! And there are so many really experienced people who comment and answer questions, willing to help others.
Two ways you can get the chickens back in the coop. For nighttime, you want to imprint the coop on them as their home and place to go at night. The way to do this is for the first several days to a week or so when you move them out to the new coop your husband is building, lock them in the coop and don't let them out in your yard. This will help them realize this is their home. Then, when you let them out for the first time, do it late-ish in the afternoon so that they only have a few hours out before it gets dark. They'll go in the coop automatically to go to sleep. Lock them in and you're done. If you do this "imprinting" process, you should never have to round them up to go to be at night. They'll put themselves to bed in the coop at dusk every night.
If you want them to go in the coop during the day, the way to do this is with treats. If you haven't already, get them used to a certain routine where you give them treats. Mine for example is this white compost type bucket I carry from the house to the barn. The chickens learned very early on that when I have that bucket in my hand, it often means there is a treat of some kind in there and they come running EVERY TIME. Even when I don't have anything for them (like, I just gave it to them out of the bucket and now I'm carrying it empty back to the house) they come running when they see that bucket. So, make it a point to give them treats with some consistent method and they'll learn it on their own. Then... when you want them to go in the coop, you use that method and in they go. Oh, a consistent chicken call helps too. I say, "here chickie chickie chickie" to call them and they come running.
In general, you don't want to have to catch them. Chasing them just scares them. You want instead, to lead them where you want them to go by teaching them to follow you. Using food! If you ever do need to pick them up, the best way to do this is at night. Just pick them up off the roost. Sometimes, you need to do this to give them medicine or check them for something. Or... if while you're giving them treats, you can grab the one you need to catch. I do this when I need to say, clean the behind of one and I want to do it during the heat of the day so the chicken will dry before dark. I'll give them treats and while they're all gathered around my feet munching away, I just lean down and pick up the one I need to work on. For whatever reason, this does NOT deter them from gathering around the next time I have treats. I guess it's just really not that often that I pick one up so they don't seem to associate treats with being picked up.
In terms of what they eat, young chicks like yours take awhile to warm up to new foods. Try cutting things up small and dropping just a few on the ground at first. For some reason, that gets them to try things. Mine eat most things, tomatoes, apple cores, bread crust, whatever scraps I have. Even carrots if I cut them up smallish. But not lettuce really. They eat enough grass and greenery since they free range so I guess they don't need any more of that. There are some good articles on what NOT to feed them (raw potato peals are a no no but cooked ones are OK, etc.). Just do a search here on maybe "best chicken treats" and you'll find the articles.
One final thought on predators... you'd be surprised (unfortunately) at how many predators will show up coming from far and wide when they think they can get a tasty chicken dinner. I free range on the edge of a national forest in southern Kentucky so we have fewer issues with suburb type animals (dogs, cats, racoons) and more real predators (coyotes, fox, hawks). I have 3 good dogs that are the ONLY reason I have not lost a chicken yet (knock on wood) to a predator. The dogs are in the house at night but outside a good deal during the day. I had to teach them to not eat the chickens themselves and once that was done, they've been awesome at predator control.
Anyway... just want to mention the predator thing again. Nothing quite so heart breaking as a dog who gets lose from some far off neighbor's yard, digs under your fence and kills your chickens you've come to love and worked so hard to nurture and raise. If nothing else, make sure you coop is super secure (hardware cloth, no chicken wire) so at least at night, you'll know they're safe when locked in there. Check out my coop (see the My Coop link under my name) for ideas on how to predator proof a coop. Loads of others have ideas and pics too, under the "coop design" forum.
Hope this helps!
Guppy