I have two browsers on my device, Chrome and Opera, but generally always want to use Chrome. But even if I select Chrome for opening weblinks and press the "Always" button, it might still ask me again which browser I want to use for some other links.
My phone is rooted (it's a Fairphone). Is it possible to just exclude Opera once and for all from being suggested? I only want it to appear when I deliberately start it via the drawer.
Unfortunately, I think not. The code that tells whether the app will appear in that list is in the app itself: Documentation.
However, you state that you are rooted, so maybe you could use Titanium Backup (Pro) to "freeze" Opera.