Plankton's last name is Plankton. His full name is Sheldon J. Plankton.
Plankton's computer is named Karen.
In "The Adhesive Duck Deficiency," his middle name is revealed as "Lee." Perhaps this is why his mother calls him "Shelly."
In the episode, "Plankton's Army" Plankton announced his full name once he got to the secret formula. He said, "I, Sheldon Jay Plankton," So his middle name must be "Jay".However, if you look at the transcript for the epison, Plankton says, "Sheldon J. Plankton." Therefore, his middle name is not mention, but his middle initial is "J."
As far as the TV show, it's Lee.
On May 18th, Cartoon Network showed an episode of Bugs, Daffy, and the two gophers. (episode 1, "Best Friends"). In it Bugs had to answer a question: What is Daffy's Middle name? Daffy held up a sign saying "Armando". Bugs said "Daffy, you know your middle name is Sheldon!" Daffy said "That is true, but 'Armando' sounds cooler." Thus Daffy's middle name is "Sheldon"
Bert Sheldon's birth name is Herbert Sheldon.
Sheldon Charron's birth name is Sheldon Charron.
Sheldon is his last name. His Real name is Robert Sheldon
Linn Sheldon's birth name is Linn Richard Sheldon.
Sheldon Lettich's birth name is Sheldon Bernard Lettich.
Blake Sheldon's birth name is Blake Fitzgerald Sheldon.