For the last decade, conductive domain walls (CDWs) in single crystals of the uniaxial model ferroelectric lithium niobate (LiNbO$_3$, LNO) have shown to reach resistances more than 10 orders of magnitude lower as compared to the surrounding bulk, with charge carriers being firmly confined to sheets of a few nanometers in width. LNO thus currently witnesses an increased attention since bearing the potential for variably designing room-temperature nanoelectronic circuits and devices based on such CDWs. In this context, the reliable determination of the fundamental transport parameters of LNO CDWs, in particular the 2D charge carrier density $n_{2D}$ and their mobility $\mu_{H}$, are of highest interests. In this contribution here, we present and apply a robust and easy-to-prepare Hall-effect measurement setup, by adapting the standard 4-probe van-der-Pauw method to contacting a single, hexagonally-shaped domain wall that fully penetrates the 200-$\mu$m-thick LNO bulk single crystal. We then determine $n_{2D}$ and $\mu_{H}$ for a broad set of external magnetic fields $B$, as well as under the variable angular dependence of the Lorentz force. Lastly, we present photo-Hall measurements of one and the same DW, by determining the impact of super-bandgap illumination on the 2D charge carrier density $n_{2D}$.