Consider the energy function $E=\frac12 x'^2+\frac14x^4$, then the time derivative of that is
The only solution that has $x'=0$ constantly is the stationary solution $x=0$. All other solutions continuously lose energy and fall towards the state $E=0$.
The second question is more complicated, you have to show that the friction is over-critical, so that no oscillation occurs. The equation $x''+x^3=0$ without friction is conservative and oscillates along the level curves of $E$, for a small friction coefficient this oscillation persists, giving a spiral in phase space.
As this plot of $x(t)$ shows, a friction coefficient of $0.8$ still crosses the zero line, any proof method for 2. must be quite specific for $c>0.9$.