Hi,
I had not ever used Ahrefs.com until I saw you post and went and tried it, and I saw that my sites had links that I did not even know existed. Ahrefs.com's database may not be updated that often and, therefore, shows you links that are no longer on the web or they are showing links that OSE has not yet picked sprawled.
Here is my best shot at answering your questions;
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Moz rank has their ranking system based on their own factors/metrics. Ahrefs.com will have their own also.
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Just-Discoverd links have a date and time on the far right of the page in the "Date Discovered (UTC)" column. I have checked out your site, and you have no just discovered links yet. I tweet and share the page that is linking to my site in all of my social accounts so the new link would show up in the "Just Discovered" section. Though this no longer works, but I feel it is good practice as having a page link to your site with more social shares and secondary links pointing to it is a better link.
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Your MOZ rank is at a 0.00 because you do not have any "popular links" linking to your site. The way to improve it and I just did a quick search and copy and pasted this for you.
"A web page's MozRank can be improved by getting lots of links from semi-popular pages or a few links from very popular pages."
Best of luck,
cibble.