Transient analysis of filters inside allowed band

I understand that a zone pass filter will cut off portions of the signal in the cutoff zone, and allow or amplify the frequency parts of the signal in the allowed zone.

Now lets assume a signal that is in the allowed zone. If we exclude minor amplifications, will the signal change in time due to all the components of the filter? I mean resistors, capacitors or inductors.

So, if we have for example a sine signal in the allowed zone, will it change (excluding the amplitude due to minor amplifications)? Or will there be some modifications? For example getting triangular a bit?

Does it make sense to try to analyze filters in the time domain? Or we just accept them as they are (black box approach) and just work with them only in the frequency domain?

Or will there be some modifications? For example getting triangular a bit?

No, not if the filter is linear. By more or less definition, the output signal from a linear filter does not contain any frequency that isn't present in the input signal.

If the input is a sinusoidal signal and the output is not a sinusoidal signal, even if by just a bit, the filter is not linear since, as Fourier analysis shows, a non-sinusoidal signal necessarily has multiple sinusoidal components of different, related frequencies.

Thus, to make the sinusoid triangular a bit requires adding frequency components that are not present in the input signal, i.e., adding harmonic distortion.

In summary, if the filter is linear, a sinusoidal input of (angular) frequency $\omega$ will result in a sinusoidal output of frequency $\omega$ with, at most, a modified amplitude and phase.

$$v_I(t) = \cos\omega t$$

$$v_O(t) = |H(\omega)|\cos[\omega t + \phi(\omega)]$$

Does it make sense to try to analyze filters in the time domain? Or we just accept them as they are (black box approach) and just work with them only in the frequency domain?

Analysing them in both the time domain and frequency domain gives more insight and, if designing a filter, it's usually best to do so in both domains. For instance, in the frequency domain it's really easy to see the frequency response but, really not easy to see how the filter performs with transient inputs likes square waves or pulses. In the time domain it's dead easy to see overshoots due to transient inputs (that may of course not be desireable) but quite a bit more difficult to see what the frequency response is - this is why we use the two domains.

It sounds like you are referring to a band-pass filter and if your input sinwave is within the pass-band area then you can totally expect it to look like a sinewave after being filtered.

"So, if we have for example a sine signal in the allowed zone, will it change (excluding the amplitude due to minor amplifications)? Or will there be some modifications? For example getting triangular a bit?"

It seems you are speaking of a bandpass filter and about a frequency that is within the "passband region" (you call it "allowed zone). If the signal is a pure sinusoidal it will not change this characteristic (the amplitude will be altered, more or less, depending on the filter gain) - however, there is only one single frequency within the passband which will not be shifted in its PHASE (if compared with the input phase): That is the center frequency of the bandpass. Other frequencies within this region (within the bandwidth of the filter) will be phase shifted. The amount of phase shift depends on the quality factor Q of the bandpass (Q: center frequency divided by the bandwidth)-