You have been getting good advice so far. While there are a few things about nests that are good to know, people tend to worry about this kind of stuff a lot more than the chickens. Chickens are really adaptable and there are a lot of different ways to do this. Your pullet’s behaviors are normal, by the way.
Adult hens will often scratch around in a nest to get the bedding just right. If the lip is too low, they can and will scratch out bedding, fake eggs, or real eggs another hen has laid. You need enough of a lip to keep that from happening. I like enough depth of bedding so the bottom of the nest stays padded too but part of that depends on what you use for bedding. We use things from feed bags, hay, straw, wood shavings, Spanish moss, shredded paper, carpet, rags, you name it. They all can work but we all have our individual preferences. I cut long grass from places I don’t mow or weed eat and dry that, so I guess that’s technically hay. Doesn’t mean its best, it’s just what I use. It’s free and I’m cheap.
I’ve seen chickens perch on some really thin edges. Will a hen hop on the edge of a milk carton to get in? It’s quite possible. Will she turn it over? Are you talking about a tiny Serama hen or a huge Jersey Giant? How firm is it where it is? Is it rickety? Some people just set a cardboard box or milk crate in the corner of the coop on the floor for full-sized hens and it works. Whether or not you need some kind of frame for the milk carton depends on your chickens and how it is set up.
Chickens don’t care that much whether the nest has a top on it or not. You can find plenty of photos on this forum where people have nests without tops. You don’t want them pooping in the nest, so don’t position it where poop from the roosts will hit it. Make your roosts higher than anything you don’t want them sleeping in or on, like the edges of your milk crates
Hens tend to lay in protected places. Their instinct is to hide a nest from predators. They tend to like darker spots. Some people have their coops so bright with lots of windows that it’s hard to find a dark spot. Covered nests tend to work better there.
Just because you provide good nests doesn’t mean they will use them. They might find a dark spot or even shadows on the floor might be enough to make them think it’s a hidden spot and will make a good nest. One of the tricks we use is to put fake eggs in the nest. They tend to like to lay where others are already laying. If there is an egg there, they tend to lay there. I use golf balls but others use wooden or ceramic eggs, plastic Easter eggs (you might fill them with sand or something heavy and glue them shut), even ping pong balls for bantams.
You’ll notice I’ve used some weasel words like “tend”. They are living animals and don’t always do what they are supposed to do. For everything I’ve mentioned, someone can come up with an exception. They can be pretty aggravating at times. But it’s a fun trip. Welcome to the adventure.