You can answer most of your questions by reading the Chrome developer pages quickly.
- "Extensions are small software that personalize the browsing experience." 
- "Chrome API" refers to the APIs used to create a Chrome extension (in addition to those available directly for web pages). 
- "By default, extensions do not work in incognito windows." 
Since the NPAPI plug-ins are no longer supported, there is no other type of "program" that you can use to modify Google Chrome, apart from distributing a modified version of Chromium.
Extensions must always be hosted on the Chrome Web Store, as loading from other sources has been disabled for security reasons.
The only exception seems to be the configuration for enterprise deployments on Windows, which allows for the installation, activation, and forced configuration of extensions. Such extensions will always be visible users, but can not be deleted or disabled without removing the domain or registry setting that installed them.
If the computer is associated with an Active Directory domain, it may be that the extension is not found on the Chrome Web Store, assuming that a managed PC in such an environment is already under the Full control of the network administrator and not the local user.