Does 1,3-dichloroallene possess a C2 symmetry axis?
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: