How many different organic structures (from the pure theoretical
viewpoint) can be drawed with only 4 (exact) carbon atoms and
with/without hydrogen?
We could make strict rules like each carbon has exactly 4 bonds and get a specific answer, but this is not reality. There can be lone pair electrons and unpaired electrons. The octet rule is not stictly followed.
$\ce {C_4}$ actually has been observed and is linear.
:C=C=C=C:
$\ce {C_4}$ has been the subject of numerous theoretical and experimetal papers because of its possible occurrence in nebulae. It has been debated whether a singlet linear, triplet linear or rhombic (kite shape) state is the lowest energy state. http://www.sciencedirect.com/science/article/pii/S0009261400005765
Neither the linear nor rhombic states follow the naive rules.
Linear $\ce{C_4H}$ has been observed both in the lab and outer space.
In fact according to the University of Kohn lists Molecules in Space, linear $\ce{C_4H}$ is one of only 61 molecules and molecular ions found in extragalactic space and of only 190 found in the interstellar medium or circumstellar shells as of 2016.
For $\ce{C_4H_2}$ linear butadiyne in known. Cyclobutatriene, cyclobutenyne and tetrahedrene have been ruled out theoretically as not represtenting any actual potential energy local minimum, while similar to $\ce {C_4}$, structures having lone pair or unpaired electrons and not following the octet rule (such as carbenes) have been calculated to represent actual minima. See the following references for theoretical cyclic $\ce{C_4H_2}$ structures:
http://onlinelibrary.wiley.com/doi/10.1002/jcc.540020211/pdf
http://pubs.acs.org/doi/pdf/10.1021/jo060698k
http://pubs.acs.org/doi/pdf/10.1021/jo000941u
In outer space, not only has the usual HCCCCH isomer been found but also $\ce{H2CCCC}$
See Observations of cumulene carbenes, H2CCCC and H2CCC, in TMC-1