<p>The first objective of this paper is to characterize all possible parameters of Plotkin-optimal two-homogeneous weight regular projective codes over finite chain rings, as well as their weight distributions. We show the existence of codes with these parameters by constructing an infinite family of two-homogeneous weight codes. The parameters of their Gray images have the same weight distribution as that of the two-weight codes of type SU1 in the sense of Calderbank and Kantor (Bull. Lond. Math. Soc., 18 (1986) 97-122). Further, we also construct three-homogeneous weight regular projective codes over finite chain rings combining with some known results. Finally, we study applications of our constructed codes in secret sharing schemes and graph theory. In particular, infinite families of strongly regular graphs and strongly walk-regular graphs with non-trivial parameters are obtained.</p>