Frequency analysis to determine low-pass cut off frequency
I think you will get almost the same amount of frequency content for both the cases (fft of 60 ms and fft of 2 sec). The time series indicates that only from 0.6 to 1.2 sec there is vibration. But it's better to use the full signal while doing the FFT as there is no fear of losing any data. If you cut some portion from the time series, you have to be careful about choosing the window (use cosine tapering not rectangular window) as edging effect may be introduced into your frequency content. 
Sorry, I didn't get the question. Could you please rephrase it?
I think that for all the trials, the time series would be different. Therefore, it is not recommended to average all the time series. It is better to analyze all the time sequences and obtain the frequency contents. If dominant frequencies for all the trials are similar, then the average frequency can be calculated. If not, you can try to find out the reason e.g. inconsistency in the sample material etc.