I think Amaymon's rather shallow - he just wants to find a good fight. The rest of the time he's bored. Mephisto was able to use that to goad Amaymon into fighting Rin in order to view Rin's full uncontrolled power.
To Mephisto, the world's a stage... and he's the playwright. Everyone is his pawn. He's taken care to control Rin, the biggest wildcard, and the Order.
He could've left Gehenna out of boredom, out of trauma, or he was kicked out. If he was kicked out, it'd be amusing that Satan wants him back. If he was upset, that... just makes me wonder, I don't believe that'd do anything to such a guy... demon... whatever.
I like to think that Mephisto is a chessmaster in this story. He doesn't appear often but it is apparent that he's carefully making his move.
Rin is his biggest (and most dangerous) pawn since we still don't know the limit of Rin's power and we know Rin is not loyal in any sense to Mephisto. If Mephisto is not careful, his own moves can attack him back.
Amaimon is a rather dangerous pawn too since if it's as you say that he's only bored, it will be kinda hard to move this pawn since it can go in unpredictable ways.
The biggest mystery is Mephisto's motivation though... You won't do such dangerous moves if its not a very serious matter.
I'm looking forward to the revelation of his past. :P
Rin is not loyal to anyone, I suppose. At the moment he seems to be "loyal" to human, but who knows, if his demon power runs out of control. I agree with what yukihime said. Thinking again, Mephisto maybe has a motive for "demon's sake", but I think he has personal reasons, private ones, that he wishes he could control Rin for himself and any purpose he has in his head.
I think he plays it right, trying to find pawns as many as possible. I think that Neu-sensei (I forgot his complete name, sorry; I meant the one who was trying to kill Rin, the one with one blind eye) is also his pawn. In general, he looks like a very careful person, indeed.
Rin is loyal to Yukio and his classmates. If Mephisto proves to be as shady as he seems, a very dangerous pawn could turn against him. Ultimately I believe his goal is to depose Satan and unite Amaimon with Gehenna.
I think Mephisto knows Amaimon's characteristics very well and use it to his own advantage. Well, to be a good chessmaster, you have to know your enemy's and your pawns' characteristics. Somehow I think Mephisto is deeper than he seems. Maybe he has layers and layers of plans in case his enemy or his pawns don't react as he predicts. *shrug* Well, who knows.
Boredom is definitely Amaimon's defining characteristic. I am assuming that is why he, of all the kings, was the perfect choice to test Rin's power and expose him.
Mephisto is definitely deeper than he seems. He does not seem like a bad guy either. Just someone with his own agenda. Definitely not a classic good guy either, at least not at this point in the story. It seems to me that everyone is his pawn, and he is manipulating a little army to achieve whatever his true goal may be.