But do only Sikh men have the last name of Singh?
Of course not. By itself, the name isn't probative. My point is that it reinforces the explicit statement that he was a Sikh. The question, one more time, is whether there was anything actually stated in the episode that suggested he was a Sikh. Of course Marla actually saying he was probably a Sikh is the main thing, but the name is worth mentioning too for the sake of thoroughness.
What about the different design for the cryogenic capsules in Into Darkness? How do we explain that?
Was it ever actually stated that the capsules seen in the film were the same ones from the Botany Bay
, as opposed to newer ones they were moved to?
If that was stated, it doesn't need any more explanation than that this is fiction and different artists render the same thing in different ways. Roddenberry himself liked to suggest that TOS was just the best approximation that they were able to manage with the available budget and resources, and that the underlying "reality" probably looked a lot better.