If information is conserved and the universe is finite, it's also cyclic, and the entropy law is false.
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The second law is about entropy of subsystems of the universe (i.e., most systems that are considered practically). Even if you have a global unitary, you have a second law of quantum complexity, which yields a second law for increased entropy of subsystems. It all checks out.
The subsystem is either isolated or not. If it's not isolated, the second law doesn't apply. If it's isolated, what I said applies and there's a contradiction with the second law.
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No, if a subsystem is isolated, then the subsystem's free Hamiltonian is the total Hamiltonian. Time evolution is governed by the Liouville equation, the Hamiltonian dynamics are unitary, the eigenvalues of the density matrix are conserved, and hence so is its Von Neumann entropy
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*Liouville-von Neumann equation
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