If I make a Progressive Web App (PWA), and I change the display in the manifest.json file to browser instead of the typical standalone preference people usually choose for PWAs, and the user opens the app and surfs to some third-party site outside my control, such as Reddit.com, does this app still have control such as the ability to add Javascript and CSS into the page kind of like a Chrome Extension would?
display: "browser" does not give you a stand-alone application. It opens in a normal browser tab/window
I'm not sure of any browser that will actually let that be installable, either.
That doesn't pass installability criteria for Chrome, for instance
As for being able to inject CSS and JS into other pages, no, you can't do that, with any display mode. It's just a website in a fancy display. PWAs aren't web views like in mobile apps, and the browser still owns the overall display context (rendering, extensions, etc…), not the installed app