What kind of carbon ions exist?
Many different carbon ions exist in the gas phase.  

In Electronic spectra of linear carbon anions, spectra of $\ce{C_2^-}$, $\ce{C_3^-}$... $\ce{C_10^-}$ are discussed.  

Dianions such as $\ce{C_7^{2-}}$ have also been observed.

In the solid phase, aluminum carbide and berylium carbide have more than 50% ionic character according to the Handbook of Refractory carbides and nitrides, and correspond to $\ce{C^{4-}}$.  Calcium carbide, which could be considered a salt of acetylene, corresponds to  $\ce{C2^{2-}}$  rather than $\ce{C^{2-}}$