I noticed that by using small window width more significant peaks appear. How can I choose the width of the window minimizing the errors of taking a false significance?
Generally, the window width is the width of your sample set. By making the width smaller than this, you are adding non-physical modulation to your signal. Windowing should always be used with care as it affects sidebands and signal magnitude. Google search yields all sorts of pages on the topic. Why not start with Wkipedia? https://en.wikipedia.org/wiki/Window_function
A longer (rectangular) window in time for the DFT/FFT will decrease the spacing between the frequency bins (f_del in cycles per sample). There is a reciprocal relationship here: f_del = 1/M, where M is the length of the window (in samples). To convert f_del to F_del (cycles per second or Hz) just multiply by the sampling frequency, i.e. F_del = F_s*f_del.
But in practice, there is more to "frequency resolution" than the bin spacing. Pure sinusoids that do not coincide with one of the bin centers will be smeared over many bins (due to the high sidelobes of the rectangular window). This may mask weaker signals, even if they are well separated in frequency. Tapered windows are therefore usually used to lower the sidelobes. But tapered windows broaden the response of each bin, so it becomes harder to distinguish sinusoids that are closely spaced in frequency.
All this is for sinusoids, that persist forever. In practice, you may be interested in transient signals. There is no point using a long window for short signals.
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