FFT returns large low-frequency power - probably because signal is interpreted as containing cycles that do not fit. How is this called? So I have a large number of signals like the one in the first picture below and I would like to extract and compare the frequencies within them. 
I applied a Fourier transform which resulted in results like those shown in the second picture. The large power of very low frequencies suggests to me that the signals include something that is interpreted as being of such low frequency that one cycle does not fit into the time window (which is 1.2 seconds).
I hence applied a High pass filter with different cut-offs and the pattern is gone after using a HPF with 2Hz (last picture).

I have two questions about this:
1) Does my reasoning make sense?
2) How is such an effect / pattern called? 

I need to include this in a paper I am currently drafting and hence am especially unsure about question 2). My supervisor's first reaction was that these are edge-effects, but looking into them I feel like that is not really what is at work here.

Thank you!!
Your reasoning sounds correct. It is impossible to see from your example what frequency that low frequency is.

If the frequency of the phenomenon is 0 Hz, then it is simply a direct current (DC) bias that affects the first bin of the discrete Fourier transform (DFT). If none of your signal of interest is in that bin then you can simply ignore it in your analyses or set it to zero. If your signal never goes to negative values, then you have DC bias (although there could be other low frequencies).