I'm not sure if this theory is used by many-worlders, but eigenstates of the Hamiltonian don't interact with each other at all. It could be that a similar selection rule explains why many-worlds eigenstates don't seem to interact.
> What causes this creation of matter and energy?
You don't need to create energy to put a quantum system in a superposition of more states. You divide the energy of each state by its amplitude squared to get the total energy, and the amplitudes squared sum to one. So in Many Worlds, as with regular QM, the amplitudes of each state get smaller as you add more states. Total energy doesn't need to change.
What is the physical explanation as to why they don't interact? I.e. does gravity not travel along that space-time dimension? Is there a second time dimension? Are the universes created infinitely far apart?
That is the physical side. Your physical intuition may be getting in the way.
> What is the embodiment of the energy that is being infinitely divided?
What is the embodiment of the energy eigenstates of a quantum harmonic oscillator in a superposition? I'm not sure what kind of answer you want, but it's the same as the answer to that.
> What is the physical explanation as to why they don't interact?
Because they're (in my hypothetical scenario) eigenstates of the Hamiltonian. To get an intuition as to why they don't interact (again, in my scenario that might not correspond to leading many-worlds theories), look at the simplest possible case; one dimensional quantum harmonic oscillators. There's no short answer to your question.
> I.e. does gravity not travel along that space-time dimension? Is there a second time dimension? Are the universes created infinitely far apart?
Unfortunately, none of these questions are in the right ballpark. They are too complicated. If a selection rule in the universal Hamiltonian (or something) is the answer, they just won't interact at all. Bringing notions of other dimensions, or great distances, or whatever into it is unnecessary.
Unfortunately, I don't think any suggestion I can give here will alleviate your confusion without having a background in QM. You may want to read Griffiths QM textbook if you're really interested.
Given direct experience of one universe, I think it's more ontologically conservative to introduce multiple universes than it is to introduce a whole new unprecedented category of cats that are simultaneously dead and alive.
Very loose analogy: if I see a mouse in front of me and simultaneously hear a squeak behind me, I'd probably assume two mice rather than one mouse which has learned ventriloquism.
Which makes it mildly ironic that John Wheeler, a leading early proponent of many-worlds, also proposed the one-electron universe...