I too was involved in a long, and ultimately only semi-successful search for a copy of Autodesk Vault Server. (Yes, Server is what you really want to do version control. The Vault version included in the Inventor download is just a client.) You can find the thread that ultimitely resulted in a Server download
here. The only problem is that Vault Server requires either a Professional or Server edition of Windows (which I don't have).
I have had similar experiences with the terrors associated with trying to "fake" version control using Dropbox or flash drives. The basic thing I learned is that top level assemblies really aren't that useful, except possibly for checking for interference between sub-assemblies. If you can live with the "Cross-part reference failure" in the top level assembly, you can probably survive without version control software. If one person can keep the only active copy of a sub-system on their computer, you can probably keep the model usable. Not to say I don't recommend backing up files, just that without proper version control, two different people really can't make edits on the same part.
A quick tip, if you aren't doing it already, use Projects in Inventor, or the SolidWorks equivalent. It really helps with keeping cross-part references and assemblies in general working coherently.
In short, if you don't want the problems you have been experiencing, you need proper version control. There are ways to mitigate the fact that you lack this, most importantly keeping one designer working on one system. It just gets too hard to try to collaborate without real version control.
By the way, I wouldn't recommend switching to Inventor just because they have Vault. SolidWorks PDM really does the same thing.