To answer my own question: Yes, this will copy any local or network folder to a Dropbox account. I haven’t found a way yet to create a new folder on Dropbox, however, so if you want to copy to a new folder, you must create it yourself on the Dropbox site or through the Dropbox sync feature.
If you got the previous version offered here, and it is now indicating inside the app that an upgrade to this version is available, you can upgrade to this version from inside the app without losing your registration info. Minor upgrades like this from version 10.x.x.x to version 10.x.x.x are free.
When it goes to version 11.x.x.x, you will have to pay to upgrade, however.