it's a tradeoff. if you put a script in your document it saves an extra request for a server resource. however an external script can be cached by the browser and reused for later visits or on other pages. an external script also reduces the size of the html document. in some cases, serving an external script from a separate domain, especially if it's on a content delivery network, can help performance. if the script is served from a cookieless domain that can also improve performance.
No, first connections anywhere are always made at a loss in performance. But, assuming you are not using some fringe library, chances are that people have common domains cached in their browser/router/DNS.