If two systems are in thermal equilibrium with a third, they are in thermal equilibrium with one another.
Agreed. The laws of thermodynamics are barely even physical laws, so much as realizations of mathematical principles in our physics. Entropy works out to a statistical property. It's not that it's impossible for it to decrease; it could happen if you are lucky enough. But low probability states have low probability, so don't come up as often by chance. Kind of stuck with that.
Conservation of energy follows from Noether's theorem. It's our physics that connects it to things like motion, but given that it is, you can't get more unless the conditions of that theorem are violated – cases like field discontinuities or where laws of motion are not quite constant over time. In fact our universe doesn't quite match them, because it is expanding, and energy isn't quite conserved, because the redshift that causes leaks some away. But it's close enough for anything on our little planet.
One of the problems with classical mechanics was how an atom could be stable; if the electrons are orbiting the nucleus, they would be accelerating and so losing energy to EM radiation. Quantum mechanics resolves that by making them standing waves. The electron is around the nucleus, but it's not really moving around it.