Cunningham is on the island when Widmore is the leader (based on his accent).
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.
I do like your theory, seeing as how Richard and Mikhail can heal, it is likely that Cunningham can heal too. But he did recur, he was in two episodes.
Cunningham is one of the original US soldiers. He didn't get killed, but joined the Others instead
Possible since Widmore referred to him as Cunningham when he told Richard what happened, as Cunningham was on his uniform (versus Widmore's uniform, which identified him as Jones)
They were posing as soldiers. He used the name on the uniform to keep up the ruse.