How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. When applied to a forced system)?
A system can be treated as a single ODE for vector-valued function. The comment-answer by Alexei Averchenko explains the concept of time-invariance and some consequences of having this structure:
Concerning the part
The implication for the force is that it's constant in time. Some such forces are conservative (have a potential function associated with it): gravitational or electrostatical. Others are not, e.g., friction. In the former case, there is a conserved "energy"; in the latter you might be able to write down a Liapunov function (a quantity that does not increase with time).