Some insects, such as beetles, are able to store their wings under their elytra by folding them and can rapidly deploy their wings for flight. The crease patterns of these wings allow the fold/unfold kinematics to take place using simple manipulation, and to remain stable in both configurations. It has been observed that the structures of the beetle wings are kinetically multi-stable origami. The crease pattern of these wings is comprised of a peculiar arrangement of four-fold vertices. In this manuscript, we show preliminary work towards studying the non-flat (conical) four-fold vertices observed in the wing structure using experiments and rigid origami analysis. We construct four-fold origami paper cones of varying angles and study their snap-through behavior under varying point-load configurations. From these experiments, the threshold forces, displacements and duration timescale of snap-through buckling are extracted. Similarly, we study the snap-through instability of two-dimensional (2D) arches having a vertex, which provide insights into the wing folds and are hypothesized to represent properties which facilitate the deployability of the wing. Using the pseudo-rigid body model (PRBM) , we numerically analyze the kinematics and potential energy of the snap-though buckling of 2D arches, and show that the model captures the kinematic behavior sufficiently well to provide insights of energetic behavior from kinematic experimental results. Overall, our approach shows promise in studying the design and kinetics of the insect wing origami, and could enable the design of bio-inspired deployable engineering structures.