With the terminal value $ξ^-$ admitting a certain exponential moment and $ξ^+$ admitting every exponential moments or being bounded, we establish several existence and uniqueness results for unbounded solutions of backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a monotonicity condition with general growth in the first unknown variable $y$ and a convexity condition with quadratic growth in the second unknown variable $z$. In particular, the generator $g$ may be not locally-Lipschitz continuous in $y$. This generalizes some results reported in \cite{Delbaen 2011} by relaxing the continuity and growth of $g$ in $y$. We also give an explicit expression of the first process in the unique unbounded solution of a BSDE when the generator $g$ is jointly convex in $(y,z)$ and has a linear growth in $y$ and a quadratic growth in $z$. Finally, we put forward the corresponding comparison theorems for unbounded solutions of the preceding BSDEs. These results are proved by those existing ideas and some innovative ones.