Sharing a Dropbox link directly to the clipboard, without having a browser pop open? Is there any solution for retrieving a Dropbox share link to a file, that doesn't involve opening a browser?

That puts it directly in the clipboard.

(For sake of clarity, I'm referring to when you right-click a file and select Dropbox → Share Link.)
No, you can't just get the link in clipboard, which is much safer for you. With Dropbox's version 2.0.5, you can easily share them through the toolbar.

There's a second way of doing it, being a lot better, which is:

Finder > Dropbox > public (move your public file here) > right click > dropbox > Copy public Link