In the casting call Cunningham is described as "Mid-20s to Early 30s. Any ethnicity. Athletic, used to living outdoors and off the land. He is a loyal soldier but willing to compromise with the enemy to protect his men. Very nice recurring character." This part (in bold) leads us to believe that Widmore did not actually kill him.
It only takes 2 appearances to be considered a recurring character and he appeared in two episodes: "The Lie" and "Jughead".
Cunningham is one of the original US soldiers. He didn't get killed, but joined the Others instead. He does have an American accent.
Possible since Widmore referred to him as "Cunningham" when he told Richard what happened, and as "Cunningham" was on his uniform (versus Widmore's uniform, which identified him as Jones, and he was clearly called "Charles").
Possible, since Cunningham seemed more easily persuaded, and less hostile than the other Others, which may imply that he was a rookie Other. Widmore appears to have murdered Cunningham very carelessly, and apart from Widmore's obvious ruthlessness, this may also be due to Widmore's opposition of allowing any of the American soldiers to join them (Widmore seems to have been very anti-outsider in his youth, though this may or may not have been diminished with age). Ellie also seemed to have been suspicious of Widmore's story, and this may be because she was aware of his contempt for Cunningham, but of course, she may have just been a little contemptuous of Widmore.
Possible because the Others seem to go by a "real" name basis (with the exception of abbreviations and such), and code names are rare and given out of necessity (Jeremy Bentham, Henry Gale, ect).
They were posing as soldiers, so he may have just used the name on the uniform to keep up the ruse.