Chiral centres with stereoisomeric substituents The structure of one particular isomer of 4-chlorohepta-2,5-diene is shown here. I am confused about how two of the propenyl groups attached to the central carbon have the same atoms with different configurations.

More specifically:

Is the carbon shown in the image is a chiral carbon or pseudo-chiral carbon?

Will this compound show optical isomerism, i.e. does it have an enantiomer?

Are the two alkenyl groups considered to be different from each other, or the same?

If the attached side centers were chiral centers (e.g $\ce{-CHClCH3}$) with different configurations, instead of geometrical isomers, then would this compound show optical isomerism?
Let's address your concerns one at a time:

The structure 1 you have drawn is chiral with $\ce{C4}$ having four different groups attached to it. $\ce{C4}$ has the (S)-configuration with the CIP priorities Cl&gt;(Z)-propenyl&gt;(E)-propenyl&gt;H.
The structure has an enantiomer of the (R)-configuration. Just switch any two groups: Cl, (Z)-propenyl, (E)-propenyl or H. Each enantiomer will have the same specific rotation but opposite in sign.
The two propenyl groups are different stereoisomers, just as different as the (Z)- and (E)-2-butenes.
If the two propenyl groups were changed to -CHClCH3, say, (2R,4R)-2,3,4-trichloropentane 2a ($\ce{CH3CHClCHClCHClCH3}$), then the central carbon, $\ce{C3}$, is chirotopic but non-stereogenic. That is to say, if the chlorine and hydrogen at $\ce{C3}$ were switched as in 2b, the compound would still be chiral but not a different stereoisomer. Rotate 2a by 180o about a vertical axis and you will obtain 2b. They are identical.