Hey man, it says that you can't answer Private Messages, so I'll post it here.
*SPOILERS SPOILERS SPOILERS*
King is never explained at all. However, I'm going to infer a few things about his past:
1. There is a scar on his face. It looks like just one of those 'tough-guy' anime scars, but there could be background to it. I suspect that he also fought against the Doctor alongside Arthur.
2. There are red flowers in the basement of Arthur's house. I believe Arthur consumed some red flowers in a last-ditch attempt to fight off the doctor, and was defeated and killed. If King fought against the Doctor as well, then there is a chance that he could have also taken a small amount of red flower (maybe Arthur stopped him from eating a whole flower?).
As far as Sue is concerned, I don't think the Doctor is her father. You meet a character later in the game named Momorin Sakamoto - she is Sue's mother, but says nothing of Sue's father.
Basically what happened was, the Doctor, Prof. Booster, and the three members of the Sakamoto family came to the island to do research. When the Doctor realised the type of power he could wield with the Demon Crown, he ditched the rest of the group. With the Demon Crown he was able to control Misery and Balrog, and used Misery to turn Sue into a Mimiga. I think your guess is correct: he wanted to test the red flowers on Sue but she got away.
Now, since the Doctor's past is not fully explained there's a definite possibility that he could be Sue's father. But I'm not too sure about that.