Object at rest or object that is moving with constant speed has same free body diagram (FBD).
How do I know looking only at FBD is object moving or at rest?
You don't. The FBD can give you the acceleration, but not much more. Suppose the acceleration is zero, then depending on your inertial frame, the object could be either moving at a constant velocity or be at rest. To conclude something about the motion based on FBD, you'd need more info, such as an initial condition.
Free body diagrams are mainly used to represent forces, acceleration, velocity etc related to the body (or system). They help us to understand the given information in a better manner.
After making the diagram, now you have to understand the situation and perform required calculations as asked by the question.
You can interpret FBDs as a snapshot of the body (system) taken at some particular time. At that time forces, acceleration, velocities related to the body is represented in the diagram and then you perform calculations.
Hope this helps