From my observations, droppers are of metal and they do not have any insulation from the catenary wire or from the contact wire.
The catenary and contact both carry the same phase and voltage so any voltage drop that occurs in the contact wire can be compensated through the catenary wire through droppers.
If droppers carry current then why do we need 'G-jumpers?'
Trolley line maintainer here.
Because, trying to insulate the catenary wire from the contact wire would be sheer madness. It's much easier to simply insulate the whole catenary-dropper-contact assembly from "the rest of the world".
Also, on lower voltage systems, line work is often done "live" (so that the line car is able to self-propel, as it is often an electric car itself. The workers on the roof platform are at the same voltage as the overhead line, so ground is lethal to them. As such, we want the entire workspace at the same potential, including both wires and the cross-arms to the extent possible.
Now, your jumper in that photo appears to be linking two contact wires and two catenary wires. That appears to be connecting two lines to each other. This is common in actively tensioned lines with counterweights, where trolley segments are segmented (each segment has its own tensioner, and the length is limited so the tensioner range doesn't have to be huge). You need to pass power from one segment to another.
