Installing both a native app and its related chrome extension in one setup
You can install a chrome extension programmatically, but you have to publish it via the chrome store to do so. You can publish the extension as unlisted, meaning it won't show up in search, but it has to be in the chrome store to do so.
The reason Google does this is so they can run malware/virus checks on the extensions to prevent malicious installation of them. To clarify from your question, it is impossible to programmatically install an extension that's not published to the chrome web store.
Documentation for this is available here for Windows: https://developer.chrome.com/extensions/external_extensions#registry
There are alternative methods on that same page for Mac/Linux as well.
Essentially, what you'll do is add a registry key that links to your extension in the web store, and handles updating/etc. This will not require any user interaction, so I think it should work for your purposes.