No its not. Period. Alexa only measures the physical traffic that comes to your website from all over the world. The more traffic, the higher the alexa rank you get. If directories can get you lots of traffic then yes Alexa will rank your website higher.
Alexa rank mainly depends on the traffic, higher traffic means good Alexa. But yes as my personal experience we I get few mentions on other authority blogs then also I observed slight boost.
No, Alexa is not like Google's Page Rank, Alexa doesn't count inbound link, outbound links etc. It just track peoples using their toolbar, extension, apps etc. Thus they create estimated number of visits a website got. No use of links their..
you can improve alexa redirection via a different domain to some sort of a low ranking. A little unethical, but it works.
Alexa rank can very easily manipulated and doesn't mean anything. Alexa only ranks better if the visitors have Alexa Toolbar otherwise not. That is why a lot of directories, SEO forums etc. rank always better than casual sites. But if you have lower Alexa it doesn't automatically mean that you have more visitors or have a better site.
might be useful in this way, try asking your friends to review and gave the rating profile of your site on Alexa.
Alexa rank is just about toolbar visits. It is not something you should be bothered about or spend time on improving it.