Linear ODE vs Linear System
Yes. In practice it may only be an approximation, and it is almost certainly only linear across a defined input/output range (e.g. consider what happens when an opamp saturates).

Well, if you put a sine wave into the system, and it only produces a sine wave at the same frequency at the output, then it is linear. If you have the equation it is trivial to determine linearity (hint, check there are no non-linear operators used on the independent variable).