Google is a .com, so it would be a .com, except I don't think Google counts it's index as a "link" at all.
If you are linked from a .gov or .edu page though it should carry some pretty good PR with the link. No one has to click the link for it to count as a backlink. Google (and other search engines) find the link when they index the page it is on.
Google does count clicks from their searchs and measures the time you spend on the page after you click. Both those factors go into determining your page rank. Your page rank determines your position in the search results. Google tries to put the most popular pages first.