It's easy - just have her log into Dropbox.com, right-click on the folder you want synced between the two of you, and click "Invite to folder". Make sure she uses the email address that your Dropbox account uses.
Is there no way to do it with only the link? Is there some kind of script or app that can scan the link for updates then download the new files?
Another question: if I manually click the download button in that Dropbox folder, how can I download straight to GoogleDrive or Box? I would want to spare my hard drive if I did this.
Not properly via dropbox. Since the share link is just a web address, you could automate the visiting of that address via a script or browser extension, though it's possible that Dropbox has some sort of security set up against that (in which case you'd need a more complex script).
Another question: if I manually click the download button in that Dropbox folder, how can I download straight to GoogleDrive or Box? I would want to spare my hard drive if I did this.
Depends on the browser. Some will let you choose where to save the file, others will place them in your Downloads folder automatically. If you don't want the latter to happen, all the major browsers will have a tickbox in the settings like in Chrome which has "Ask where to save each file before downloading".