Application of the Sampling Theorem to a System
How do I apply the sampling theorem to a system so I can choose the
  correct sampling period?


You can't apply a sampling regime to a transfer function of a system because all the TF contains are placements for poles and zeroes and it contains no knowledge about the input to the system in the frequency domain.

You could make an argument that the TF is basically a low pass filter and that beyond a certain input frequency the output will be very small but there's nothing in your question that allows this judgement call to be made.


  Furthermore if I use a wrong sampling frequency how can I demonstrate
  that there is something wrong with the system?


Engineers decide beforehand what the input frequency limits are and apply anti-alias filters to prevent significant aliasing errors so, if something comes along that is larger than expected in normal situations and might cause an aliasing error then you could compare the RMS of the anti aliasing filter output with the raw signal and make a decision based on this.