FFT frequency bands and filtering
Regarding the transition band of the lowpass and highpass filters; an ideal filter would indeed have zero transition bandwidth but it would also have infintely many filter coefficients in its impulse response. In practice you will have a finite length approximation to that infinite length impulse response, as a result the practical filer will have non-zero transition bandwidth. In case of an FIR (finite impulse response) implementation, the longer the impulse response the narrower will be the transition bandwidth.  As a result of this nonzero transition band of the finite length FIR filter, the resulting frequency response will display deviations from the ideal response through the transition band, you should (and can) take this into account. You can also consider using IIR based recursive filters, but that will still have a transition band, in additon to the nonlinear phase distortion being introduced, which doesn't exist for a typical (symmetric) linear phase FIR filter.

Regarding your second question on the spectral analysis of the filtered signal through its FFT, you first obtain a long enough signal observation window and then take a dense enough FFT of the filter output, so that you can focus on any region withn 1 Hz to 40 Hz depending on your interest. Assuming that your sampling rate is $F_s$ samples per second, then an $N$-point FFT would provide you an equivalent analog frequency resolution of $$\Delta f = \frac{F_s}{N}$$ where each frequency bin, in Hz, would be located at $$f_k = \frac{F_s}{N} k ~~,~~ \text{for}~ k=0,1,...,N-1$$ Note that FFT of real valued input signal would have conjugate symmetric complex output, so only the first half (roughly) carries relevant information from $f=0$ to $f=F_s/2$

Note that your spectral resolution is fundamentally limited by your observation window length i.e., FFT length does not improve frequency resolution, but merely interpolates the already resolved spectrum.