  In this article, we study the diameter two properties (D2Ps), the diametral
diameter two properties (diametral D2Ps), and the Daugavet property in
Orlicz-Lorentz spaces equipped with the Luxemburg norm. First, we characterize
the Radon-Nikod\'ym property of Orlicz-Lorentz spaces in full generality by
considering all finite real-valued Orlicz functions. To show this, the
fundamental functions of their K\"othe dual spaces defined by extended
real-valued Orlicz functions are computed. We also show that if an Orlicz
function does not satisfy the appropriate $\Delta_2$-condition, the
Orlicz-Lorentz space and its order-continuous subspace have the strong diameter
two property. Consequently, given that an Orlicz function is an N-function at
infinity, the same condition characterizes the diameter two properties of
Orlicz-Lorentz spaces as well as the octahedralities of their K\"othe dual
spaces. The Orlicz-Lorentz function spaces with the Daugavet property and the
diametral D2Ps are isometrically isomorphic to $L_1$ when the weight function
is regular. In the process, we observe that every locally uniformly nonsquare
point is not a $\Delta$-point. This fact provides another class of real Banach
spaces without $\Delta$-points. As another application, it is shown that for
Orlicz-Lorentz spaces equipped with the Luxemburg norm defined by an N-function
at infinity, their K\"othe dual spaces do not have the local diameter two
property, and so as other (diametral) diameter two properties and the Daugavet
property.
