I leave all applications in the Applications folder (some are in the Utilities subfolder), then I create my own folders to organize subsets of them to suit my purposes. I copy aliases of the real apps into these folders. So I don't have two copies, but I can have just the sets of apps I want. And I can drag any of these folders into the dock or Finder sidebar.
You can make an alias of an application by selecting its .app file and then using File->Make Alias, or by dragging the .app file to another folder while holding command and option.
Some applications have a folder in the Applications folder instead of a single .app file. For those, open the folder and find the actual application.
One other advantage to using aliases of applications: you can rename them. For example, the web browser application is named Firefox.app, but in my folder of applications I can call it just Firefox, or maybe FF.