Ignoring band-limiting, filtering and such:
A perfect square-wave has only two levels. If those two levels happens to fall on quantizer levels (1 bit would do), the quantization noise would be exactly zero. SNR is infinite.
Yes, if the noise is suitably sized compare to the quantizer levels. SNR would depend on the (shot) noise in the signal and in the (read) noise introduced by the electronics, even at 1 bit - see for instance 1-bit ADCs in audio.
Standard engineering approaches to discrete sampling assumes that signal and quantization error is uncorrelated and one or both uniformly distributed, thus the quantization "noise" can be calculated independent of signal. This leads to the SNR=6.02*#bits formula.
Even ignoring what was said above, given suitably sized sample and noise I wonder if this applies in imaging where the signal is not a slowly changing, repeating sine wave.
Dither is the willful introduction of more noise, usually with the intent of encoding more (low-frequency) levels. Noiseshaping can move this noise into frequencies where it is less annoying.
Yes, although 'proper' dithering can also be provided by (read and shot) noise present in the system, without having to add any. That seems to be the case in modern DSLRs, where input referred read noise tends to be around 1 ADU. Witness for instance very similar engineering DR as measured at 12 and 14 bits for the same camera:
12.9 vs 13.2 stops resp. for the D7000, for example.