We have used molecular dynamics simulations for a comprehensive study of phase separation in a two-dimensional single component off-lattice model where particles interact through the Lennard-Jones potential. Via state-of-the-art methods we have analyzed simulation data on structure, growth and aging for nonequilibrium evolutions in the model. These data were obtained following quenches of well-equilibrated homogeneous configurations, with density close to the critical value, to various temperatures inside the miscibility gap, having vapor-liquid as well as vapor-solid coexistence. For the vapor-liquid phase separation we observe that $ell$, the average domain length, grows with time ($t$) as $t^{1/2}$, a behavior that has connection with hydrodynamics. At low enough temperature, a sharp crossover of this time dependence to a much slower, temperature dependent, growth is identified within the time scale of our simulations, implying solid-like final state of the high density phase. This crossover is, interestingly, accompanied by strong differences in domain morphology and other structural aspects between the two situations. For aging, we have presented results for the order-parameter autocorrelation function. This quantity exhibits data-collapse with respect to $ell/ell_w$, $ell$ and $ell_w$ being the average domain lengths at times $t$ and $t_w$ ($leq t$), respectively, the latter being the age of a system. Corresponding scaling function follows a power-law decay: $~sim (ell/ell_w)^{-lambda}$, for $tgg t_w$. The decay exponent $lambda$, for the vapor-liquid case, is accurately estimated via the application of an advanced finite-size scaling method. The obtained value is observed to satisfy a bound.
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