I think with the Mad King, the emphasis should be on "Mad" -- he behaved destructively because he was mentally impaired / had some kind of actual disorganized thinking pattern. True psychotic break.
I agree about Joffrey, who was just kind of some sadist; he liked having power, he liked using his power, and he had no real feelings for those he used his power against, so he would be as prone to hurt or spare someone just to see what happened as he would be to torment them. it was like bug-crushing. However, he wasn't impervious: There were a few moments where his pride had been hurt, however, and it revealed he could be touched -- and these were typically scenes in which Tyrion had bested/humiliated him somehow publicly. Tyrion was not only the one who it seemed that Joffrey could not rattle or harm even when he tried, but who had disdain/indifference towards Joffreys' power and would put him in his place.
I'm not sure what is up with Littlefinger myself. So far it's come across that he loved Cat but when she married Rob, he basically just gave up on love completely and invested his energy into politics since little mattered to him anymore. it does seem like a 'game' to him, and he's invested in playing it even as basically just to see how far he can actually get and perhaps even win.