Separate Sookie for a start, I think is the best bet. If you don't yet have an extra cage, well, this is one of the reasons everyone needs one.
They are a great investment and you will always need one at random times, often when you least expected it.
Getting a new cage and putting her in there every night for a week to bond her to it as her sleeping place can possibly solve the social issues, but won't solve the egg destroying issues. Because of the way she's hurting the other hens, I'd lock her away for a week or so instead of just letting her free range with them and changing her roosting place at night. I'm going to suggest a few things which might be a better bet, but given that she's displaying multiple negative behaviors, treatment is not so simple.
She will soon train the other hens to eat the eggs, in all likelihood. It's necessary to train her out of egg eating and scratching the nest up. She definitely lacks maternal instinct to even go digging in the nest like that, and that's likely how she learned how to gain an egg meal. But this doesn't mean it's guaranteed she'll never be a mother if you choose to let her mother, just that it's unlikely.
Separating Sookie for a week or so will rearrange her on the social order. All the other hens will take a step up and Sookie will fall to the very bottom. This may allow other bullies to emerge, as sometimes subordinates become nasty when enabled to become dominants. They were often subordinates in the first place because they were not fit for the dominant role. But sometimes the dominant bird is only so because they are excessively nasty.
When Sookie is returned she will have to reestablish herself, which does likely mean fronting up to each hen in turn and possible fights. I would first take nail-clippers, the sort used for people, and trim off the see-through tip of her upper beak. I'd file it until it's smooth after trimming off the excess. This won't hurt, unless you cut into the opaque part. It will let her eat, drink, and preen normally, but smashing her beak hard into an egg to break it will hurt and she won't do that. It will also prevent her from being quite as damaging to other hens, but it will grow back rather fast.
Otherwise, if you choose not to trim her beak, then I would not give her a nest box in the new cage unless it's one that has the special flooring to roll the egg away from her safely. The pattern of destroying nests and receiving rewards of raw egg for it must be broken. I would suggest getting sand for her nest and if possible, no walls, or deep walls, so she has more trouble smashing it.
When you return her, it might help to shackle her feet, if you think she may take the fights too far and be vicious rather than fair, which sounds fairly likely. I shackle their feet by taking a soft, round, thick shoelace, and tying it very securely around each ankle, leaving enough room to take a normal step, but not enough to run. It's important to make sure it won't slip over either ankle or knee, and that the knots won't slip and tighten. Like this, she can dustbathe, walk normally, jump up and down from the perch, scratch, etc, but if another chicken tries to escape her, she can't chase it. I would provide no distraction when she is reintroduced, since that means it's more likely the other hens will all be focusing on her. The shackles can be cut off with scissors any time you think she's learned her lesson. Easy to do on the perch at night.
Inherent risks in separating any chicken, even for a few hours, can include: subordinates stepping up into violent dictators (if that was already in their natures, not that you'd ever see it if they never got the chance to step up); total rejection by the flock; permanent outsider or low status, and extra violent status battles. None of this might happen but how a flock reacts to having the social order forcibly changed is unpredictable.
Best wishes.
