I found some handouts on the subject, but I am in doubt about the following
Composition property. The evolution from the time $t_0$ to a later time $t_2$ should be equivalent to the evolution from the initial time $t_0$ to an intermediate time $t_1$ followed by the evolution from $t_1$ to the final time $t_2$, i.e. $$U(t_2, t_0) = U(t_2, t_1)U(t_1, t_0) \qquad (t_2 > t_1 > t_0)$$
Is this always true, even for a time-dependent Hamiltonian $\hat H(t)$?
Yes, it is always true. If you evolve a state vector $v$ present at time $t_0$ to a later time $t_1$, then you'll have $$ U(t_1,t_0)v $$ Then, if you evolve that vector state from $t_1$ to $t_2$, then you'll have $$ U(t_2,t_1)U(t_1,t_0)v, $$ which had better be the same as evolving $v$ from $t_0$ to $t_2$: $$ U(t_2,t_0)v = U(t_2,t_1)U(t_1,t_0)v $$ Time evolution has this 'exponential' property. If the system is time independent, then the evolution depends only on the difference between times, and that is a straight exponential property: $$ U(t_2-t_1)U(t_1-t_0)=U(t_2-t_0). $$