None of my files will open now: "The project seems to be of an older format,..."

You might want to check with Dropbox support.

My best guess would be that you have “smart” syncing turned on, which causes the Dropbox app to only download files as needed. That’s known to cause big problems for Scrivener.

Downloading through your browser forces Dropbox to give you the whole directory – a Scrivener project is actually a folder, with subfolders and potentially hundreds of component files – which is what you want.