The majority of those pushing matches after the whistle are a continuation of the play from before it. Defender is trying to clear the front of the net, offensive player is trying to secure position to screen the goalie, get a deflection, or tuck away a rebound. When the whistle blows the players are still tangled and the pushing continues. Sometimes its the offensive player who took an extra whack or two at the puck after it was blown dead that draws the push. These hits are very superficial, they are mainly a "I saw what you did, knock it off" message.
There is plenty of sportsmanship in the NHL. Almost every fight is between willing participants who discuss it first to ensure the other party isn't playing with an injury or something. Handshake lines after playoff series when you just spent the past week and a half playing the same opponent every other day. There is plenty of shit talking from the benches, but wingers will typically shoot the shit a bit when lining up for faceoffs.
Why is fighting only a 5 minute penalty? Because see above about "dem da rules". Fighting has always been a part of the game, and has fallen dramatically in the past few years. Players don't tend to get head injuries from fights, and when they do its because they fell awkwardly and hit the ice with their head. The hits that are borderline dirty or overly aggressive just aren't there, knowing that you might have to put up for a fight for giving the hit. Those hits are the ones that are causing the injuries. From behind, sending your opponent face first into the boards. Lining up the big hit from across the ice. Steve Moore jumping on Tood Bertuzzi's back and driving his head into the ice. Those are the plays that have lasting affects.