Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. One major problem concerns the stability of steady states of these networks. Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steady-state that yields periodic oscillations)? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm -- our Newton-polytope method -- gives an easy-to-check condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Another important invariant of a chemical reaction network is its maximum number of steady states. This number, however, is in general difficult to compute, as it translates to counting positive real solutions of parametrized polynomial systems. To this end, we introduce an upper bound on this number -- namely, a network's mixed volume -- that is easy to compute. In this talk, we apply our two new tools to an important biological-signaling network, called the ERK network. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and periodic oscillations (for a very particular choice of initial conditions). Conradi and Shiu (2015) proved that when certain reactions are omitted, the ERK network reduces to the processive dual-site phosphorylation network, which has a unique, stable steady-state (for any initial conditions). This stark contrast in dynamics prompted Rubinstein et al.'s question, "How are bistability and oscillations lost as reactions are removed from the ERK network?" By analyzing subnetworks of the ERK network, we systematically answer this question and demonstrate that bistability and oscillations persist even after we greatly simplify the model (by making reactions irreversible and removing intermediate species).