Are there any mathematically conceivable arrangements of this universe (particularly ones that respect its mass-energy content) that cannot in principle be achieved through its evolution in accordance with the known natural laws, but must remain forever out of reach (assuming that the universe is indeterministic and that the mass-energy content of the universe is always the same)? Also, according to the current understanding of quantum mechanics, is it physically possible for the current state of the universe to lead to another state and change back to the same state, and then lead to another totally different state only to change back to the same identical state again? If this is physically possible, then it would seem to me that an indeterministic account of the universe must also allow for the world to evolve and cycle back via endless routes.
The answer to your first question depends on what you mean by "mathematically conceivable". If your definition of "mathematically conceivable" is wide enough then, yes, there will certainly be "mathematically conceivable" states that are not physically possible. If your definition of "mathematically conceivable" is "consistent with the laws of physics as we understand them" then, by definition, a "mathematically conceivable" state must be physically possible.
Can a given physically possible state be reached by evolution from some initial state ? Yes, trivially, if we take the initial state to be the same as the target state. Can a given physically possible state be reached by evolution from an initial state that falls within a restricted set of initial states ? The answer to that depends on how you decide to restrict the initial states.
Would any known laws of physics be broken if the universe cycled endlessly through a fixed series of states ? No, they would not - although it might be inconceivably unlikely. Cosmologist Roger Penrose has proposed a cyclic model of the evolution of the universe known as conformal cyclic cosmology.