Do Maxwell's equations contain any information on the time evolution of the current density $J$? The answers to Can the Lorentz force expression be derived from Maxwell's equations? make clear that Maxwell's equations contain only information on the evolution of the fields, and not their effects upon charges; the Lorentz force equation is an added equation.
Does this imply that any arbitrary time evolution of a current density can be defined beforehand, and the corresponding fields always found that satisfy Maxwell's equations?
 A: Maxwell's equations place a constraint on the current, namely that it be conserved. To see this, take the divergence of Ampere's law for
$$0 = \mu_0 \nabla \cdot \mathbf{J} + \mu_0 \epsilon_0 \nabla \cdot \frac{\partial \mathbf{E}}{\partial t}$$
which is equivalent to
$$\nabla \cdot \mathbf{J} = - \epsilon_0 \frac{\partial}{\partial t} (\nabla \cdot \mathbf{E}) = - \frac{\partial \rho}{\partial t}.$$
This is precisely the statement of charge conservation. If you plug in a $\rho(\mathbf{r}, t)$ and $\mathbf{J}(\mathbf{r}, t)$ that aren't conserved, then the equations will have no solutions at all.
A: 
Does this imply that any arbitrary time evolution of current density can be defined beforehand, and the corresponding fields always found that satisfy Maxwell's equations?

Yes, given a charge density $\rho(\mathbf r,t)$ and a current density $\mathbf J(\mathbf r,t)$, you can find fields $\mathbf E(\mathbf r,t)$ and $\mathbf B(\mathbf r,t)$ satisfying Maxwell’s equations.
See Wikipedia for the integrals giving the scalar potential $\varphi$ and vector potential $\mathbf A$ that solve the nonhomogeneous wave equations with sources $\rho$ and $\mathbf J$. The fields derived from these potentials will satisfy Maxwell’s equations.
One way to think about this is that an arbitrary charge and current density can be considered a swarm of moving point charges. The fields of an arbitrarily moving point charge is known, based on the Liénard-Wiechert potentials. The fields of the swarm are simply the superposition of the fields of all the point charges, by the linearity of Maxwell’s equations.
ADDENDUM: As @knzhou points out in another answer, the $\rho$ and $\mathbf J$ can’t be completely arbitrary. They have to satisfy the physical constraint of current conservation, $\partial\rho/\partial t+\nabla\cdot\mathbf J=0$.
