Nuada's fighting style is mostly based on speed and agility, doing all kinds of elaborate maneuvers to land hits on his opponent without being hit back. And he shares a magical link with his twin sister that means that everything that physically happens to him also happens to her and vice/versa. It stands to reason that he developed his fighting style in order to keep his sister from getting hurt along with him, supported by the first fight scene where he gets hit in the face and then turns to Nuala, clearly horrified as they both start bleeding.
Hellboy's Kind Hearted Cat Lover traits make sense due to his hellish origins. Cats are believed to be the gatekeepers of the underworld.
It's entirely possible Hellboy didn't prevent the apocalypse, he just assured that it would happen when it was meant to. He barely aged a year after over 300 years in hell, and he's in his 60s to 80s, so the apocalypse likely isn't meant to happen for millenia. This also implies he COULD have been the anti-christ if he hadn't been summoned to earth.
When Liz goes nuclear, everything around her burns. If she steps in water, it boils. Yet, every time, her clothes go untouched.
It's magic fire. Hell, it's Vril fire, the flamma reconditus itself. It doesn't have to follow the rules of regular fire.
I've always wondered: when Hellboy gets pushed under the train while trying to get to Sammael, he gets his horn stubs hit often enough to cause heat friction. He touches the stumps and says "ouch". But then one scene later he electrocutes Sammael and says, "I'm fireproof, you're not." Not a huge issue, but always kind of bothered me.