Domain authority is basically nothing to do with Rich Snippets Directly
However some rich snippets work indirectly because they may only display when your within a certain rank on the page and some rich snippets will only display for certain keywords. So if you don't rank for "socks" but your ranked top for "buy socks Bournemouth" then the likelihood is that the rich snippets will not be displayed because of them not being enabled for this term.
Domain authority comes naturally when you start building great content and gain virtual PR through visits, back links, social mentions, and age... the list goes on and on... Your best off
looking at ways to increase a site position in Google. Everyone wants to gain domain authority and it's not a simple here 10's things.. It's over a book's worth, but with this said Pro Webmasters has tons of SEO questions and answers, some of the best threads can be found by searching for tags and view in seo best votes for example.