Dropbox doesn't access all your files, it just hooks to explorer.exe and then whatever directory you open in explorer Dropbox will query info about that directory, files and subdirectories. To test this, just close every explorer window, run Total Commander and Process Monitor, and use Total Commander to do your stuff, you will see that Dropbox won't be quering directories/files.

And about that connections to dropbox servers, it needs them to sync files if they were uploaded from another computer/smartphone.