Yes. For example: A square wave has a Fourier series.
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Yes it can.
yes it can, if you know how to use or have mathematica have a look at this demo http://demonstrations.wolfram.com/ApproximationOfDiscontinuousFunctionsByFourierSeries/
An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.
Fourier series is the sum of sinusoids representing the given function which has to be analysed whereas discrete fourier transform is a function which we get when summation is done.
A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.