Hi! I'm sorry you're in a rough situation--it sounds like you need to have a real talk with your partner about what you both want and where you see things going. The cutesy nickname could be really indicative of something, or it could just be a bad habit. My ex and I tried to stop using the nicknames we had during our relationship, but we fell back into it. It's affectionate, but not a sign that we have any romantic interest in each other anymore. But we don't consider each other best friends, and only are in touch a few times a year. I can see how it could be a sign that they aren't really done with each other yet, even after all this time. And I'd be inclined to be more worried if they are in contact a lot and about what else they are saying, you know?
Living together or apart is a personal choice, but again, I think you should be on the same page about it, and find out why she wants to keep her own place if she's spending most of her time at yours. It may be a really good reason (or at least an understandable one) or it may not be. Once you know you'll be in a better place to make a decision about what you want for yourself and your family.
I think normal is hard--everyone works out their own situations based on what they need and want. But if you aren't happy, you have a right to figure out what will make you happy, and if your partner can't (or won't) work to give you that (if you are doing the same for her) then it might not be a relationship worth sustaining. Communicating is key, though--she can't change anything if she doesn't know what's wrong, you know? Good luck!
And so are the boys!