Is the violation of time-reversal always associated with the violation of time-translation and vice-versa? Is the violation of time-reversal symmetry always associated with the violation of time-translation symmetry? What about the converse? Is it possible for one to be violated while the other remaining conserved?
 A: 
Is the violation of time-reversal symmetry always associated with the violation of time-translation symmetry?

If we are willing to consider non-Minkowski spacetimes, then the answer is definitely no. The Kerr metric for a rotating black hole is invariant under time translations but not under time reversal. Time reversal changes the sign of its spin. 
What if we only consider Minkowski spacetime? At least if we're willing to consider quantum field theory, then I think the answer is still no. Nominally, the Standard Model of particle physics has time-translation symmetry but not time-reversal symmetry. I say "nominally" because (last time I checked) we still do not have a mathematically rigorous nonperturbative definition of the Standard Model — or of any nonabelian chiral gauge theory, as far as I know, not even on a lattice. However, if we delete the gauge fields, then I think the no-gauge-field version can be rigorously constructed on a discrete spatial lattice in a Hamiltonian formulation with continuous time-translation symmetry, and I think it still doesn't have time-reversal symmetry. 

What about the converse?

If $t$ is any given point in time, let $R(t)$ denote the time-reversal through  $t$. Then the product $R(t)R(t')$ is a time-translation by an amount $2(t'-t)$. Therefore, time-reversal symmetry about all times would imply time-translation symmetry, because translations can be expressed as pairs of reflections.
However, symmetry under time-reversal about a single point in time does not imply time-translation symmetry. If we are willing to consider arbitrary non-Minkowski spacetimes, then we can contrive an example of a (classical or quantum) model that has time-reversal symmetry about $t=0$ but that does not have time-translation symmetry just by contriving a metric field with this property and then constructing a (classical or quantum) field theory in this prescribed background.
Even if we consider only models defined in Minkowski spacetime, then we can still contrive examples by, say, considering a (classical or quantum) field theory with a prescribed classical background scalar field $\phi$ that has time-reversal symmetry about $t=0$ but not time-translation symmetry, such as $\phi(t) = \tanh^2(t)$.
If we don't want to include any time-dependent backgrounds at all, then I think we're stuck with time-translation symmetry by definition.
