How come the flatness problem still a problem while there is an era of cosmic accelertion? The equation  
$\Omega-1=\frac{k}{\dot{a}^2}$
together with the assumption that $\ddot{a}<0$
are the equations which demonstrates the flatness problem " because currently $\Omega$ is so close to unity and they mean that at early Universe $\Omega$ was closer to unity because $\dot{a}$ is a decreasing function" . 
How come that we ignore that currently there is a cosmic acceleration where $\ddot{a}>0$?! ain't it possible that there is no fine-tuning of initial conditions and we just fall into an almost flat Universe?
 A: The universe has not been accelerating for long enough to make that much difference to the flatness problem.
The Friedmann acceleration equation is
$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left( \rho + \frac{3P}{c^2}\right)$$
where $\rho = \rho_M + \rho_{\Lambda}$, with $\rho_M$ the density of matter (and radiation), whilst $\rho_\Lambda$ is the energy density of the vacuum. For the standard cosmological constant $P = -\rho_{\Lambda} c^2$, so
$$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\left( \rho_M - 2\rho_{\Lambda}\right).$$
Thus the acceleration only becomes positive when $\rho_M < 2\rho_{\Lambda}$ or when $\Omega_M < 2\Omega_{\Lambda}$. 
At present $\Omega_M \sim 0.3$ and $\Omega_{\Lambda}\sim 0.7$ so the expansion is accelerating. However, as recently as $\sim 6$ billion years ago, the universe was decelerating, would have had $\Omega \sim 1$ and thus $\dot{a}$ would have been larger before that. At those earlier times then dark energy density would have been even less important compared with matter density and so the assumption that $\ddot{a}<0$ would have been even more correct and hence the flatness problem remains.
