It occurred to me that during gravitational lensing, the light curves around a given object, creating a curved path. Because the path of light is curved, and thus, longer than a straight path, would that affect the age at which we see the object whose light's being curved?
Yes it does. Thus if a background lensed source is varying in time, then this variability can be seen at different times in the variously lensed images with a distinct lag between them that is related to the optical path length of the lensed light.
Such time delays can be hundreds of days and have been measured in the optical and radio light from quasars. The time lag offers a means of determining the Hubble parameter (e.g. Tewes et al. 2013).