Is frequency for dc zero Hz?

Very clever, but that's not how it works.

By your reasoning you should not only be able to make the frequency infinite, but also 4 Hz, or 100 Hz, or \$\sqrt{2}\$ Hz, all at the same time, with the same signal. And that's why you can't do that: a repeating signal can have only 1 fundamental frequency, which is 1/period.

It would be the same as taking 2 periods of the 4 Hz sine and saying that that's the period, because it also repeats, and then the signal would be 2 Hz. It can't be 2 Hz and 4 Hz at the same time.

Yes you can treat an infinite line as a repeating segment of some arbitrary wavelength to obtain a periodic signal. However, the function within this period is a flat zero. So if we look into the frequency domain of this periodic signal, we will see that it has no amplitude at its fundamental, nor any harmonics. They are all zero. If you like, you can pretend that the signal is of some frequency, any frequency you like, but zero amplitude.

Sampling any input waveform at a particular rate N will yield a result which the amplitude of any frequency component f will be the sum of the amplitudes of all frequency components kN+f and kN-f for all integer k. Thus, when sampling at rate N, a DC component will be indistinguishable from AC components at frequencies (2k+1)N/2. Note that if one samples a signal twice at frequencies whose ratio is not a rational number (say 1.0 and π), the first sample by itself would be unable to distinguish between DC and integer multiples of 1.0Hz, while the second could be unable to distinguish between DC and integer multiples of πHz. Since the only "frequency" which is an integer multiple of both 1.0Hz and πHz is 0, there is nothing other than DC which would yield a constant voltage on both samples.