Can carbon be the central atom of a complex? Can it form a quadruple bond with itself?
Asking multiple unrelated questions at once is considered a bad practice, but let's forget that for a moment.

Your first molecule has more or less satisfactory electronic configuration, but that's not enough for the long-term survival. N would be more than willing to relinquish a proton, and C to accept it, and thus in no time you will have $\ce{H2N-CH2-NH2}$, which then may or may not transform into something else.

Your second molecule has been discussed here before on multiple occasions. Long story short, it does exist, but doesn't have a quadruple bond. (Such bonds are known, but not in $\ce{C2}$; they have a delta component, and that requires d orbitals). The problem with your description is that the sigma-oriented 2p orbital of one carbon will interact not just with the similar 2p, but also with the 2s orbital of the other. If not for this fact, it would work out pretty much as you expected.