Does the Time Evolution Operator Commute with any Other Operators? Does the time evolution operator in quantum mechanics commute with any other operators, with a commutator of zero? Also, what exactly is the utility of the time evolution operator, is it more convenient than simply writing the time dependent portion explicitly?
 A: Yes.  For a given system, there can be lots of operators that commute with the time evolution operator.
The simplest example is the hamiltonian itself.  Recall that the time evolution operator for a time interval of length $t$ is
\begin{align}
  U(t) = e^{-i Ht/\hbar}
\end{align}
Now, notice that we can expand the exponential in a power series in powers of the hamiltonian, so we find that the commutator of the time-evolution operator with the Hamiltonian vanishes because the Hamiltonian commutes with every term in the series expansion;
\begin{align}
  [U(t),H] = \sum_{k=0}^\infty \frac{1}{k!}\left(\frac{it}{\hbar}\right)^k [H^k,H] = 0
\end{align}
In fact, any operator $A$ that commutes with the Hamiltonian will also automatically commute with the time evolution operator by exactly the same argument.  If such an operator is an observable, then this observable is what we, in quantum, call a conserved quantity.
In practice, the time evolution operator is useful for lots of reasons.  Here are a couple.  For one thing, it's conceptually useful.  Knowing that time-evolution in quantum mechanics occurs via the action of a unitary operator is super fundamental and is related to the "conservation of probability."  See, for example, this physics.SE post:
Relation between unitarity and conservation of probability
Also, formulating time-evolution in terms of the time-evolution operator is a very natural thing to do from the perspective of time-dependent perturbation theory when formulating it in terms of the Dyson series.
