What does the universe expand into?
I always like the easy questions...! The universe expands into nothing at all. Note that this is different from a vacuum. The universe expands into where there is nothing there. When the universe was created, spacetime, the "construct" in which the universe exists, was created with it. And spacetime is expanding with the universe. Spacetime could be seen as being created on the "wavefront" that is the outer edge of the expanding universe. There!
Of course, we could question whether the Universe is expanding. The evidence is poor. Could be.