Take a look at this example code. In it, the author is manipulating a Wavefront OBJ (3D) using Cocos2d. It's a bit overkill for what you're doing, but if you spend some time with it, you ought to be able to manipulate a 3D cube with whatever you want drawn on a surface of the cube (e.g., your level).
A much simpler solution would be to go with one of the existing scene transitions built in to Cocos2d. Take a look at CCTransitionFlipX. It's not exactly what you're asking for, but it's very similar and might be satisfactory for you. Here's a demonstration of the scene transitions from github.
To some extent, you're trying to fit a square peg in a round hole. (Or maybe I should've said, you're trying to fit a 3D cube in a 2D circle). Cocos2D is not really intended as a 3D platform, although because of the limitations of Cocos3D, you're not the first to attempt to try workarounds in Cocos2D. I think it's worth seriously considering switching engines to one designed for 3D (like Cocos3D, as you mentioned). Once you have one 3D element, you're bound to want more.
You asked, "How will I have the other part of the level in place while the camera rotates/rotating?" This is a fairly broad question. Obviously you will have to do some (or all) of the loading of the next level before you can render it to a surface of a 3D object and render it to the scene.
Finally, it is indeed possible to use Box2D with the Cocos3D engine. Here's a link to a demo that implements Box2D in Cocos3D. Here's another. There aren't a ton of examples or discussion on how to do this, so you're going to be struggling with some issues to make this work, but you might find this page helpful, where someone shared a few of their lessons attempting to combine the two.
Hope this helps.