You have that the wrong way round. PageRank and Alexa Rank are metrics used to assess/predict the popularity and quality of a page. (See my answer to your other topic for more information.)
In what? If you mean in search engine ranking, then they have no role. They are metrics created and used by moz.com to estimate how well a domain or page might perform in search results. (As a page’s performance might differ quite widely for different search terms, it’s hard to see how they can be any real guide.)
The link for Domain Authority seems to be broken; I’m only seeing a blank page. https://moz.com/learn/seo/domain-authority