Is it correct to use "how's" as short for "how does"?

Solution 1:

You seem to be making the false assumption that colloquialisms are not grammatically correct. Contractions are a natural part of spoken language in any language in the world—it’s part of the very fundament of speaking that we contract things that provide little necessary information. That does not make them ungrammatical in any way; quite the opposite, in fact.

A better question to ask would be when and where it is appropriate to contract ‘how does’ to ‘how’s’.

As you yourself say, it is exceedingly common to do so in normal, informal, spoken language, and I would say there is no reason whatsoever not to do it in a colloquial situation, apart from wishing to add a particular feel, effect, emphasis, etc. to what you’re saying. In other words, it is quite on par with contracting ‘how is’ into ‘how’s’.

The more formal you get, the more likely it will become that ‘how’s’ (whether for ‘how does’ or ‘how is’) will be seen as unduly informal and not appropriate to the situation. And in writing that’s not meant to reproduce spoken language specifically, I would avoid it entirely.