1. Google does not use GA data for rankings. They only use data they have access to "outside" of your analytics, such as possibly;
- Click through rate
- bounce rate/time on site
- +1's
- "Block this site" (sometimes it shows up in the SERPs when you return)
- Google Toolbar activity like bookmarking
2. So following from question 1, it would not help for SEO to do this. It might help you however in gaining some more insight as to how logged in users use your site.
3. Its thought sub-domains do not pass much link juice across sub-domains, but for a subdomain to be on a private part of the site, this may not matter quite so much. If you've always done it that way, it may be better to just keep it the same. In these cases I always do what makes the most sense to the user. Eliminate confusion. Have a short but description word for the subdomain.
4. Compete's official documentation is here - and Rand did an interview with them back in 2008 here
Hope that all helps!!
-Dan