Umm, you can drag a box with your mouse over all files if you like.
Yes, on the desktop versions, which simply synchronise a directory.
The Android version lacks this obviously desirable functionality. I know not of the IOS version, but since the whole thing pretends not to have a filesystem in the first place, I'd be extremely surprised if it was any different.
The official Android app seems rather limited, but there are a few alternatives, which let you copy stuff to and from Dropbox. Some them may allow bulk copying or syncing automatically.
I use Ghost Commander, which is a proper file manager. It easily allows you to copy a whole folder to or from Dropbox (or Google Drive, or a shared folder on my own computer etc).
Or DropSync - not tried it myself, but it should automatically sync a directory with Dropbox.