Is OS X doing some work after moving .app to trash?
OS X doesn't contain an build-in automatic uninstaller. So it doesn't do "some" work.

AFAIR one exception is (OS X) Server.app. If you move or trash the app all services will be stopped. Even then all accompanying files and folders won't be uninstalled or deleted. 

Often apps are self-containing: all files and folders necessary to run the app are in the app.bundle itself. After trashing the app the only remaining files are preferences, cache files and sometime application support files all residing in your or the system's Library folder.

Some apps coming as an Installer.pkg are installed with the Apple Installer.app. Other apps are using a custom installer. Some of them also contain an uninstaller which usually removes all installed files.

A small number of apps come with an somehow "automatic" uninstaller e.g. MagicPrefs: after deleting/trashing the Prefpane file the app uninstalls itself.

Spotify probably installs a launch agent or launch daemon which relies on a file in the Spotify app bundle. After deleting Spotify, the agent/daemon misses its target - the "SpotifyWebHelper" - and as a result the process won't be started. The plist (and some other Spotify related files) is/are probably still there and you have to remove them manually to "clean" your system.



After installing Spotify it indeed installs a launch agent com.spotify.webhelper.plist in ~/Library/LaunchAgents:

&lt;?xml version="1.0" encoding="UTF-8"?&gt;
&lt;!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"&gt;
&lt;plist version="1.0"&gt;
&lt;dict&gt;
    &lt;key&gt;Label&lt;/key&gt;
    &lt;string&gt;com.spotify.webhelper&lt;/string&gt;
    &lt;key&gt;KeepAlive&lt;/key&gt;
    &lt;dict&gt;
    &lt;key&gt;NetworkState&lt;/key&gt;
    &lt;true/&gt;
    &lt;/dict&gt;
    &lt;key&gt;RunAtLoad&lt;/key&gt;
    &lt;true/&gt;
    &lt;key&gt;Program&lt;/key&gt;
    &lt;string&gt;/Users/user_name/Library/Application Support/Spotify/SpotifyWebHelper&lt;/string&gt;
    &lt;key&gt;SpotifyPath&lt;/key&gt;
    &lt;string&gt;/Applications/Spotify.app&lt;/string&gt;&lt;/dict&gt;
&lt;/plist&gt;


The executable /Users/user_name/Library/Application Support/Spotify/SpotifyWebHelper relies on the existence of /Applications/Spotify.app. After deleting Spotify.app it won't start. 

To remove all remains after deleting Spotify remove the launch agent in ~/Library/LaunchAgents/, the folder Spotify in ~/Library/Application Support/, two prefs files in ~/Library/Preferences/ and several folders in ~/Library/Caches/