Is my browser sending any information about my linux distribution?
Most probably. Click this link to see what your browser sends to the web server. The OS should be listed under the first category called "User Agent"
Another good site that will show you what a website can learn about your system when visiting the site is BrowserSpy.dk; the Electronic Frontier Foundation (EFF) site listed in Keltari's answer uses some of the code from BrowserSpy.dk. If you click on Browser on the menu on the left of the home page, you will see information regarding your browser. When I visit the page using Firefox on a Ubuntu Linux system, I see "Mozilla/5.0 (X11; Ubuntu; Linux i686; rv:24.0) Gecko/201001 Firefox/24.0)"