Is Cullen right-handed or left? I apparently need to know!
Everyone is right-handed in videogames.
They can rarely afford to put the time and resources necessary into making unique reversed animations for one single left-handed character, when those resources are necessary everywhere else.
That said, you could headcanon Cullen as left-handed if you wanted. I'm left-handed myself, but my dominant sword hand is still the right one, because that's just how you're trained. Sparring with your left hand against a right handed opponent means you're both sword-to-sword and shield-to-shield, and it's just a big mess that gets nowhere. And then, from using your right hand to do one thing you get into the habit of using it for others, and before you know it you've become ambidextrous (eg. playing chess with your right hand because why not).
So Cullen could easily have been naturally left-handed, then trained to use his right hand to fight, and ended up comfortable using either hand for various different little things.