No it's definitely not a normal behavior and even if i do understand the frustration some people experience about PW it is by far an excuse to act like a total buffoon and insult people for playing a deck they don't like or even hate. Also everyone that supports this kind of behavior on this forum or anywhere else for that matter should rethink their moral views. I for one don't like PW and even tho i do get salty about it i would never allow myself to get into a position to insult someone that does something i wouldn't do especially if what he or she is doing is not wrong by any means. So don't even question this at all and everyone that says this behavior is normal just proves how insane they are and are probably the same people that do the same thing
No one said you should act like it but it's perfectly normal in Hearthstone to expect the kind of response the OP mentioned if you make or accept a friend request immediately after a game. This is why I leave a cooling off period before accepting.
Nothing abnormal here, it's perfectly normal especially if you invite immediately after the game.