How to use Runge–Kutta methods in a second order ODE
I think there is a mistake in your post: the first equation should be $\ddot{x}(t) = F(t, \dot{x}(t), x(t))$ (and not $\ddot{x}(t + dt)$).
Also, in the computation of $kv_i$ and $kx_i$, it should be $kv_i = v(t) + ka_{i-1}\Delta t$ and $kx_i = x(t) + kv_i\Delta t$ (where $\Delta t = dt$ or $\frac{dt}{2}$). $kv_i$ does not use $kv_{i-1}$ (except in the computation of $ka_{i-1}$), and same thing for $kx_i$.
Question 1: in the differential equation $\ddot{x}(t) = F(t, \dot{x}(t), x(t))$, if you consider the vector $Y(t) = \begin{pmatrix}x(t)\\\dot{x}(t)\end{pmatrix}$, then you have $\dot{Y}(t) = \begin{pmatrix}\dot{x}(t)\\\ddot{x}(t)\end{pmatrix} = \begin{pmatrix}\dot{x}(t)\\F(t, \dot{x}(t), x(t))\end{pmatrix} = \tilde{F}(t, Y(t)) $. That explains why the method works (since the formulas will be the same as with a first order differential equation). Actually, the approximation should be better with the corrections I proposed.
Question 2: With your formulas, a bit of simplification gets you $v(t+dt) = v(t) + \frac{dt}{6}\left(5a(t) + 2ka_1 + 2ka_2\right)$ which is not coherent.
Even with the corrections I proposed, it is incorrect. My guess is that you should compute $x(t+dt)$ and $\dot{x}(t+dt)$ as a whole (as in the vector $Y(t)$) instead of trying to compute them one after the other. The system $\left\{\begin{array}{rcl}\dot{x}(t+dt) &amp;= &amp;\dot{x}(t) + \ddot{x}(t)dt\\ x(t+dt) &amp;=&amp;x(t) + \dot{x}(t)dt\end{array}\right.$ can be seen as $Y(t+dt) = Y(t) + \dot{Y}(t)dt$.