This study focuses on a multidimensional backward stochastic differential equation (BSDE) with a general random terminal time $τ$ taking values in $[0,+\infty]$. The generator $g$ satisfies a stochastic monotonicity condition in the first unknown variable $y$ and a stochastic Lipschitz continuity condition in the second unknown variable $z$, and it can have a more general growth with respect to $y$ than the classical one stated in (H5) of \cite{Briand2003}. Without imposing any restriction of finite moment on the stochastic coefficients, we establish a general existence and uniqueness result for the weighted solution of such BSDE in a proper weighted $L^2$-space with a suitable weighted factor. This result is proved via some innovative ideas and delicate analytical techniques, and it unifies and strengthens some existing works on BSDEs with stochastic monotonicity generators, BSDEs with stochastic Lipschitz generators, and BSDEs with deterministic Lipschitz/monotonicity generators. Then, a continuous dependence property and a stability theorem for the weighted $L^2$-solutions are given. We also derive the nonlinear Feynman-Kac formulas for both parabolic and elliptic PDEs in our context.