He killed Nathan because he (Goodwin) was arguing with The Others over Ana Lucia's being worthy (of being on the list) and he knew that had she cut off his fingers, she would never be considered and as a consequence he would lose her.
Knowledge of the Arrow Station
His reaction when the Tailies come across the Arrow station seems to signify that he did not know of its existence because a radio was just lying there and he did not want them to have it. If the Others knew of the Arrow station, they would have taken the radio already to make sure nobody else could get it. This might suggest that the Others do not know as much about the Island and the DHARMA complex as suggested.
If you listen to him, he did know about the Arrow. He speculated (correctly) that it must have been used for supply. He clearly did not want them to have a working radio and probably didn't know about the material in the box at the Arrow.
Seems likely The Others had used it for storage. The glass eye may have belonged to Mikhail, the Bible with the DHARMA tape was left over or placed there. Goodwin may have not anticipated the radio, thus offered to try and get it to work, when in reality he was planning to dispose of it. He was exposed before he could do so.
He might have been guessing. On the other hand, he might have known about the Arrow, but never been there.
Goodwin may have a background in Chemistry as opposed to the Peace Corps, as Ben set him to work at The Tempest.
He is Karl's father. Karl says that he and Alex used to lay in his backyard and name the stars. Therefore, Karl is most likely the son of one of the Others. Goodwin is a likely candidate, since he is about the right age to have a teenage son, and with him being dead, there is no longer anyone looking out for Karl.
We found out in "The Other Woman" that Goodwin was married to Harper, so if Karl was his son, then Harper would likely be Karl's mom. So, he would have someone to take care of him.
Karl's surname was revealed to be Martin, not Stanhope. If he was indeed the son of an Other, it's likely not to be Goodwin for this reason.