According to the commentary on the director's cut, DD killing Quesada in the beginning was meant to show how low he'd sunk. (His arc, as Mr B said.) In the script he knocks Quesada onto the tracks in self defense - Quesada did have a gun to his head - and the train barrels down on him too fast to save him. DD didn't mean to kill him, but he also didn't feel that bad about it. Apparently the studio liked the cold-blooded killing, and there wasn't enough in the budget to shoot it differently, or so the director says. He didn't want to show DD murdering a guy, but his hands were tied.
As for DD throwing Bullseye out the window, Bullseye had just killed his girlfriend, plus he was trying to kill DD. Plus, if you watch the little scene in the middle of the credits, Bullseye's still alive.
I dunno why I'm always defending this movie. It is quite stupid. But dammit, it's a Daredevil movie! Hollywood actually made a Daredevil movie. It's hard not to love a movie featuring one of your favorite fictional characters, especialy one you never dreamed would get a movie,even if it does suck.