The eggshell from Sherwin Williams is what I prefer to use on walls, but they might call it satin? I forget. It's good, and washes well.
Re the primer debate - I suspect there must be a product difference. Because if you use the paint I have available on a bare surface, then you're not going to get a good result, and it's not going to last as long. And most of the people I see who say they never use primer live outside the US. So there must be a difference in products.
Zolotiyeruki - I don't know why you're refinishing the floors first. By doing that, first, you have to wait for the floors to cure before you can work. Second, you'll be using ladders, etc, which could scuff the new finish. Third, you'll have to be really, really careful about covering the floors when you're painting. Accidents happen. If you're going to be sanding the floors anyway, then you can relax quite a bit about drips.
Definitely pull the carpet after you've done the ceilings at a minimum. I would actually suggest doing ceiling, trim, walls, then pulling carpet and touching up/finishing trim where it was covered by carpet previously. That way, you can use the carpet as a drop cloth.
Sounds like the kids aren't little. Make them help. I was painting by 10 for sure. I wouldn't have them do the ceiling because they'll fall over into the paint, but they absolutely can help with cleaning, moving things, painting within their reach, patching/sanding small spots, etc. Painting is one of the entry level DIY skills. Get them started.