Is Batman a murderer if he does not help someone and leaves them to die?

Not a murderer, that would go against his most defining character trait, but potentially a killer.

Legally speaking, that's manslaughter. It's against his code nonetheless.

I'm going to personally disagree with the last part. I think what a lot of writers have gotten wrong about Bruce as a character over the years is reducing his psychology to a moral code; a lot of superheroes have a generic no-killing rule out of morality, but I never thought of Bruce as having one. If he did, he would not be actively torturing his opponents, or wasting them for life physically and mentally, since that is oppressing the human life every bit as much. But he never had qualms about that. I think Bruce simply doesn't want to see people die, no matter what. It goes as far back to him losing his parents in front of him. It's more a desperate need than a moral code, so abandoning a villain to fend for themselves with very high chances of non-survival is a more brutal side of the Batman for certain, but not one that's out of the realm of his defined character, in my opinion.

Of course it's a subjective interpretation with the history of his writing, but you know.

On a personal level, i would not consider that murder.

Legaly speaking, if you deny first aid help for someone in front of you, it is considered a crime. Like when someone is hit by a driver and the driver run without giving aid. This person could be sued with an attempt of murder.

But, i'm guessing blond guy got in this situation after engaging in battle against Batman and attempting against his life himself. Batman is also injured and one could argue he could not give blondie proper help as in this situation he would need to focus his own weel being and dragging a person around the snow would most likely kill them both.

But of course, we know he COULD have donne it if he wanted, since he is freaking Batman. And he knowing that still Choses to leave the guy behind, what would actually make him guilty. But no trial in the world would be able to condem him for that, as the alegation depend of him knowing he could save the guy and himself, while he could simply state he was not able to do it.

Yep IMHO.

If he's the one that caused the injuries or lead that person to their death then of course. Murder/Killing/manslaughter call it what you want. He's the one that put that person in a state that would result in their death.

It's manslaughter, not murder

Yep, big murder