Just an update for anybody that may stumble on this in the future, and also for the devs in case they choose to look into this later on.
I managed to figure it out and solve the issue. It was the "Use hardware acceleration when available" option. Turning it off removed all the stuttering issues. It also maybe explains why when the browser is not in focus the stuttering goes away (because hardware acceleration goes to the program in focus instead).
I don't know why it's like this though. Firefox uses hardware acceleration just fine, and my hardware should be more than sufficient to run a browser.
Anyway hopefully this will help shed some light. At least now I can finally start using this otherwise awesome browser.
You could launch a command prompt and run:
"C:\Program Files\Opera\launcher.exe" --makedefaultbrowser
If that doesn't work, you can right-click the command prompt shortcut, run it as administrator and then do the same command.