As per your side note: I don't think it's flexibility, because you are better about it when you set-up. I think it is a function of strength and focus. As my buddy 2-SCOOPS is constantly telling me and everybody else "your upper back is weak and fixing that is the key to making everything better". It's harder to stay tight the deeper you get into the squat. This would explain why everything seems fine when you unrack it and then you lose that tightness when you descend (and especially on the way up).
You're correct that going a little wider than used to can make those knees cave. Just keep pushing them out hard. I really like squatting wide as it lets me keep my torso more upright and I can handle more weight, but it is a delicate balance between how much my hips can open up, how strong my glutes are to keep my knees out, and just how much of a beating my hip joints can take. Don't go too wide too fast. Just ease it out if that is your plan.