Truncating the time-domain recordings to be the lengths of the shortest recording is preferable if they all have the same sampling frequency. If they have different sampling frequencies, you will have to resample them to the same sampling frequency, because otherwise the fft will have different frequency resolutions.
Using the ensemble average of the fft of biological signals is a common technique in biomedical engineering. See for example Demonstration of Fourier Transforms: Signal Avergaing. I would simply use the regular fft. Designing a digital filter to filter baseline drift and high-frequency noise before you do the fft is certainly an option, providing you use the same filter on all signals.
