At 5000 reputation, you gain access to the site analytics pages (see the help center docs, or the site analytics page). This doesn't answer all of your questions, but it does show metrics on posts (questions, answers), votes (accept, up, down), and traffic (page views, visits, new visits), along with referring sites, search engines, and direct traffic.
Some of the information you were asking about:
- 91% of our traffic comes from searches. 7% is referrals. Only 2% is direct.
- Of the referring sites, most are Stack Exchange sites. Academia, Politics, Electronics, Puzzling, Software Engineering, and Workplace are all listed. Some of these make sense for cross-linking, but this also includes where people were when they clicked on a Hot Network Question.
- Google is the most popular search engine, by far.
I'd have to pull the CSVs into Excel to do some more analysis, but it looks like we've been average around a dozen answers and half a dozen questions since we entered public beta on February 14, 2011.
It looks like somewhere around 5-6k page views has been our weekly trend as well.
The number of new visitors has been tapering off since a peak in mid-September.