We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator $g$ allowing a general growth in the state variable $y$ and a sub-quadratic growth in the state variable $z$, like $|z|^α$ for some $α\in (1,2)$, when the terminal condition satisfies a sub-exponential moment integrability condition like $\exp\left(μL^{2/α^*}\right)$ for the conjugate $α^*$ of $α$ and a positive parameter $μ>μ_0$ with a certain value $μ_0$, which is clearly weaker than the usual $\exp(μL)$ integrability and stronger than $L^p\ (p>1)$ integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under the additional assumptions that the terminal conditions have sub-exponential moments of any order and the generators are convex or concave in $(y,z)$. Afterwards, we extend the uniqueness and comparison theorem to the non-convexity and non-concavity case, and establish a general stability result for the unbounded solutions of the preceding BSDEs. Finally, with these tools in hands, we derive the nonlinear Feynman-Kac formula in this context.