Wow, just wow. While I don't think it's the best way to handle code control, it's not entirely unusual I worked at a company with 6 developers and no source control was used, they kind of had their own internal way of managing files, a release manager and whatnot would oversee all changes. In-fact the mantra was that new functionality could be added in to projects as long as some kind of switch was wrapped around the functionality.
For example we were working on a b grade social network written in PHP and the client wanted functionality to be able to subscribe to a users profile. Basically the functionality was wrapped in a check like this if ( ENABLE_SUBSCRIBE_FUNCTIONALITY == true ) { then execute the code }
The place I worked at as well never had more than one developer at a time working on a particular file, mostly everything was modular so in that instance there was no need for source control. However, I believe the advantages of source control far outweigh the disadvantages of not having it in most cases.
The very fact your company is resorting to spreadsheets documenting file changes and what was changed when something like Git or Subversion can handle that for you is just absolutely ridiculous.