Does 1,3-dichloroallene possess a C2 symmetry axis? Is there an $C_2$ axis of symmetry in $\ce{Cl-CH=C=CH-Cl}$?
  


A $C_2$ axis of symmetry means that I should be able to rotate the molecule about an axis by $180^\circ$. But I would need a composition of the following two rotations


$90^\circ$ rotation about $\ce{C=C=C}$ axis
$180^\circ$ about the perpendicular bisector of $\ce{C=C=C}$.


Would that count as a $C_2$ axis of rotation?
A $C_2$ axis is indeed here—although not the one that you are thinking of: your suggestion of a $C_4$ then $C_2$ rotation does not qualify as a $C_2$ symmetry operation. There is a true $C_2$ axis which bisects the allene and is tilted by $45^\circ$ from both the $\ce{Cl-C-Cl}$ and $\ce{H-C-H}$ planes.
One can visualize symmetry elements for small molecules using, for example, Jmol.
Launch the program GUI and do the following:

File → Get MOL
Type in an identifier, e.g. the name, 1,3-dichloropropa-1,2-diene, or CAS number, 83682-32-0.
Open console (File → Console) and enter show pointgroup to list symmetry elements and draw pointgroup to add them to the picture: