I will be using this answer by Bob Bee to address this part of the question:
If information is conserved, why did the early universe had lower entropy than now?
It seems that the concept of conservation of information arises within a quantum mechanical system of complete solution of the early universe . This presupposes that quantization of gravity has been achieved. The quantum level does not have a definition of entropy as defined by thermodynamics, which is a classical theory, emergent from the underlying classical statistical mechanics level.
information in its basic simplest form, in quantum theory, is the state of the system (which could be composed of many subsystems). A physical system is defined by a state vector. It could and often is infinite dimensional, but could also have finite dimensional Hilbert subspaces (like the spin). The evolution of a system,considered a pure state, is given by a unitary operator which preserves causality (at the Hilbert space level, not in the probabilistic interpretation of collapse and measurements). You can always go back by applying the inverse operator. When the state becomes mixed information can be considered to be lost, and entropy increases.
The way I understand it, during the time of the universe when conservation of information holds due to a pure quantum mechanical solution for the universe, entropy is constant. Once decoherence sets in, the classical statistical mechanics, entropy increases. In this view there is no conflict between a low entropy in the beginning of the universe and fixed at a given value, and the increase after quantum mechanical solutions decohere. The entropy law has a larger or equal sign in front.
"since information is equal to entropy" is not true in the quantum level. Look at the article on information entropy which uses classical probabilities, not quantum mechanical. Also this on thermodynamics and information theory does not involve the unitarity argument . It seems that the unitarity argument is important for conservation of information in quantum systems, but not in defining information entropy.
One should also keep in mind that quantization of gravity is still an open research field.