The phrases in use are "how are you doing?" (with the 're contraction, usually, since it's a conventional phrase), and "what's new?" or "what's the news?". I've never heard "how's your news?" said by a native speaker (I just checked your pointer to the wiki article: that's a very specific use that doesn't extend to ordinary speech and sounds strange even as a professional question).
In the phrase "what's the news?", is is used because "news" is what's known as a collective singular. Collective singulars have a plural form, but are seen as an aggregate, thus taking the singular verb.
English doesn't use "new" as a noun, it's always an adjective in singular form. Saying "what's the new?" where "new" is a noun (as opposed to "what's new?", where new is an adjective), sounds very strange and unnatural.