That was only after he thought Spock had killed his people. Before that, he just wanted to kick Marcus's ass and get his people back.I really didn't see Khan as out for revenge though? He seemed more interested in recovering his people. At least that was my interpretation of events. YMMV.
Pretty much. Khan/Harrison wanted his people to be recovered, but he also wanted revenge on Marcus and Starfleet/Earth afterward, so he staged the crash of the ship into the city.
All I can say is, I think STID maybe presents the 'villains' on more of a par with each other, whereas the older movies were definitely a "primary villain, secondary villain" format. The Duras sisters took down the Ent-D, but when I think of the 'villain' in GENS I tend to overlook them completely, so relatively insignificant are they to the thrust of the plot. That movie actually feels like it's only got one villain... when it's really got three! 
