Do I need to register my site to get Alexa stats?

No.

There is no need to register your site for Alexa to report traffic statistics for it. We report on all sites for which we have the data. We get our data from a variety of sources that contribute to our global traffic panel. Based on that, we are able to create a comprehensive big-picture view of the web as a whole. If we have data on your site from our panel, we will report it. If not, we will display a notation of “No data,” which means that in the last three months, none of our panel users visited your site.

If you would like your site included in Alexa’s top sites by category lists, you should have it included in the Open Directory at http://www.dmoz.org/. Our categories are a modified copy of theirs, and updates there will be reflected at Alexa, too. Many other sites use the Open Directory, too, so including your site there is a good idea in general.