Q1 The DFT is basically the DTFT of a periodic signal. As you say, in the real world we have limited time sequences and can't deal with nonperiodic signals that extend for all time. If we have a limited time-slice, we can't even know if the signal is periodic or zero for other time values. Hence, we can implement a DFT and just assume the signal is periodic, which leads to discrete frequencies, rather than a continuum, which is also efficient in practice. Then you can decide whether your sequence should be padded with zeros around it before making it periodic, and in this way approximate a nonperiodic signal.