We study the following system of equations when , and is a sequence of general nonlinearities. The nonlocal operator is given by for a sequence of even, nonnegative and measurable jump kernels . We prove a Poincaré inequality for stable solutions of the above system for a general jump kernel . In particular, for the case of scalar equations, that is when , it reads for any and for some nonnegative and . This is a counterpart of the celebrated inequality derived by Sternberg and Zumbrun in [46] for semilinear elliptic equations that is used extensively in the literature to establish De Giorgi type results, to study phase transitions and to prove regularity properties. We then apply this inequality to finite range jump processes and to jump processes with decays to prove De Giorgi type results in two dimensions. In addition, we show that whenever or then Liouville theorems hold for each in one and two dimensions. Lastly, we provide certain energy estimates under various assumptions on the jump kernel and a Liouville theorem for the quotient of partial derivatives of u.