In the first part of the present study, Discrete Element Method (DEM) is adopted as a numerical simulation technique for studying gravity packing of randomly distributed monodispersed particles in a box of a rectangular cross section (can be thought as a fluidized bed). Packing density, coordination number distribution and radial distribution function (RDF) are calculated. Stability of the packing, spatial and temporal effects of the wall on packings are analyzed. Qualitatively and quantitatively, the results agree well with the existing literatures. Since this model uses structural reconstruction, many of the features of random packing like clear second peak split in the RDF plot have been observed. From experiments, it is well known that the confining walls impart some order in the near-wall regions. However to the best of the author's knowledge, the actual symmetries of these orderings (of walls) have never been analyzed and these have been the focus of the second part of this study. Taking a cue from structural analysis of amorphous (glassy) atomic systems, the Honeycutt-Andersen (HA) index and Bond Order Orientation (BOO) order have been employed to study the local symmetries of both near wall and core regions. It shows that the 1551 HA index, signifying icosahedral order, is more predominate in the core part than the wall. There is also some good amount of cubic symmetries both in the wall and core regions. However the most predominate structure is distorted icosahedra, which is probably appearing because of a competing effect between the icosahedral order and cubic symmetries.