If you quit a project and then close Scrivener, there will be no project that was open when Scrivener was last closed for Scrivener to reopen.
So as a test, run a project and just quit Scrivener while the project is still open. If you then relaunch Scrivener, the project should open (so long as it is still accessible to Scrivener, i.e. it hasn’t been moved or deleted) .
Open the project, do something in it, wait a few seconds so Scrivener saves it, then quit Scrivener without first closing the project. Just Command-Q. The re-open Scrivener.
Does Scrivener still open without any project open?
I was going to suggest resetting the preferences or just toggling the options off and then on again to make sure nothing is stuck.
Also noticed that Dropbox is beavering away during the opening and closing of the project, so perhaps it is locking the project in some way. Worth creating a test project on your desktop to see if that works properly, just to eliminate Dropbox as a possible cause?