Flatness puzzle in a $\Lambda$-dominated Universe Friedmann equation together with the acceleration equation yield
$\dot{\Omega} = (1+3w) H \Omega (\Omega-1)$. 
The Flatness puzzle lies in the fact that $1+3w>0$, but how is this possible in a $\Lambda$-dominated era? Shouldn’t $w=-1$?
 A: $\Omega$ is the ratio of the energy density of the Universe to the critical density, $\Omega = \frac{\rho}{\rho_c}$, where $\rho_c = \frac{3H^2}{8\pi G}$.
The flatness problem (as I understand it) is the question why the value of $\Omega$ is so close to 1 today. The flatness problem is solved by allowing an inflationary epoch in the very early Universe.
Note that $\Omega = 1$ is a trivial solution to your conservation equation $\dot{\Omega} = (1 +3w) H \Omega (\Omega - 1)$ and $w$ can hence take any value.
The condition for cosmic acceleration (not related to flatness) is given by $1 + 3w < 0$. 
This comes from the equation $\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}(\rho + 3P)$, which we can rewrite in terms of the equation of state using $w = \frac{P}{\rho}$. We know we live in a dark energy dominated era, so $\ddot{a} > 0$, and hence acceleration occurs for $(\rho + 3P) > 0$, or $1 + 3w < 0$. With some rearranging we see that $-\frac{1}{3} > w$. So a $\Lambda$ dominated epoch where $w = -1$ satisfies the acceleration condition.
