Does #"C"_2# exist, and if so, describe its electron distribution?

Actually, diatomic carbon, or #C_2#, does exist, but only as a gas and at extremely high temperatures.

This article from the literature discusses it:
http://pubs.acs.org/doi/abs/10.1021/ja00194a042

(You can view that in full if you have an ACS subscription.)

You can also prove this by constructing carbon's molecular orbital diagram.

The #C_2# molecule has a total of 8 valence electrons, 4 from each carbon atom. As you can see, 6 valence electrons occupy bonding orbitals - #sigma_(2s)^(2)#, #pi_(2px)^(2)#, and #pi_(2py)^(2)#, and only 2 occupy an antibonding orbital - #sigma_(2s)^(star 2)#.

The diatomic carbon molecule has a bond order equal to

#"B.O." = 1/2("bonding electrons"-"antibonding electrons")#

#"B.O." = 1/2(6-2) = 2#

This suggests that the two carbon atoms are bonded via a double bond. However, as you can see, it's not a "classical double bond", meaning that it is not comprised of a sigma and a pi bond

If you go by this explanation, the Lewis structure of diatomic carbon looks like:

#:"C"="C":#

Those lone pairs of electrons are highly reactive, which is why diatomic carbon can only exist at temperatures close to #4000^@"C"#.

The literature article above describes how computational chemistry was able to predict the structure of #C_2#, which would be a mixture of the #""^1 Sigma_g# and #""^3 Pi_g# states. These energy states are similar in energy #(E_(""^3 Pi_g) > E_(""^1 Sigma_g))#, so thermal excitation would allow both to exist more-or-less at the same time.

The ground-state singlet would look like a diradical #C_2# with a triple bond:

#cdot"C"-="C"cdot#

And the ground-state triplet would have two electrons in carbon 1's bonding #p#-orbital lobe and one electron each in carbon 2's bonding and antibonding #p#-orbital lobes.

(the Lewis structure of the standard prediction of diatomic carbon above IS this structure, but doesn't indicate which lobes are being occupied)