When calculating "antenna element length" should we use the speed of light in the "medium between transmitter and receiver (antenna)" or the speed of light in the "element material"? (for example speed of light in air vs speed of light in Aluminum)
The speed of EM propagation in a metal antenna is not the same as c in free space, it is always slower. This requires that the physical length of a dipole antenna be shortened, compared to the free-space wavelength of the signal to which it is tuned.
This is true for the feedline between the antenna and the transmitter as well, and when a tuned feedline is used to optimize antenna performance, this slower velocity must be taken into account, and the feedline length be shortened accordingly.
Both effects can be described in the antenna world in terms of something called the velocity factor, which is the percentage by which c is reduced in a feedline or antenna. That factor is also the percentage by which the antenna's physical length must be shortened to match up with the incident waves.
There are two effects, and they're tricky.
Dielectric material in the antenna's vicinity slows down the light in that material. But, since we usually don't have an antenna fully embedded in dielectric material, that means that the speed of light is different at different places, complicating the analysis.
The field at the ends of antenna elements isn't simple. The ends effectively have capacitance, which makes their "electrical length" longer than their physical length. If you like, you may consider this a concentration of displacement current that extends the physical current beyond the ends.
Note that radio waves don't propagate in metals, so at radio frequencies there is no speed of light in aluminum. A wire may guide an external radio wave, but the wave isn't present inside the wire.