Do you consider it even remotely possible that Stannis becomes the Night’s King?

I’d put the chances at pretty low. For one thing, from everything we’ve seen, you need to be in a place to be affected by its spirit, and Stannis hasn’t set foot in the Nightfort nor does he look likely to. Rather, Stannis is probably going to set up his HQ in Winterfell itself. 

For another, it just seems to run counter to Stannis’ whole story arc. For him to be suddenly seduced by the power of the Others when everything else has been pointing to Stannis burning away his soul in service to R’hllor seems out of nowhere and rendering a lot of previous chapters moot.

Finally, to the extent that the show has any correspondence to the books, the Night King and Stannis are two different people with different agendas.