No. Exactly the opposite. The universe expanding means that at large (ie universe) scale, energy is NOT conserved. Based on our current theories, for this expansion to occur “new energy” must be added to the universe – we call it dark energy. Dark energy is just a placeholder name – we don’t know what it is but if our current theories are correct, then there must be some kind of energy that causes the expansion. Since this adds to the total energy in the universe – conservation of energy is violated (again this is in VERY large scales, bigger than galaxies kind of scale.) In any kind of human level physics, it is still convenient to use the principle of conservation of energy to predict outcomes.
But to your first question – we have very little idea what keeps it going. It is one of those things where we believe we can measure it but have no explanation for it. If you want to dig a little deeper – different measures of the expansion rate appear to give different results (kinda hard to measure on galactic scales so these are indirect measures). We call this the Hubble Tension. (ie we are tense about the fact that different methods give different measures of the rate of expansion.. /jk) Resolving this discrepancy is on the cutting edges on the physics of cosmology today.
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