Can I say:
It opens the browser on the URL [X]
meaning that something is opening the browser with the URL [X] already typed in and loading?
Preposition on may be acceptable in that context, but is less suitable than with, using, or at. You might also reword the sentence, for more-proper semantics:
It opens the browser to display the page at URL ...
Yes, you may. With or at are more common, but a Google search for "browser on url" shows plenty of hits with this usage, for example: How to open a browser on a given URL
You may open url in/with a browser.
Order of the action: usually you open a browser, then the browser opens url.
In case you have some "shorcuts" on your "desktop" you may open "the browser with the url" by clicking it. That means, the browser will be opened with the url adress embedded in shortcut.
You may "open url in a browser" - you type the url to the browser address bar.
Personally, I have never heard of open browser on url.
I'd say it opens the browser to the url.
The url represents a location(on a server somewhere) that the browser moves to. The browser does not contain/possess the url so you can't open a browser with a url.
It's like: "Drive your car with your house." vs "Drive your car to/at your house."