Kalman Filter with Accelerometer with DC Offset
Q1: Without actually implementing it and seeing that it works, your formulation looks pretty good... except that I do not understand your final equation:
$$
z_k=\left[\begin{array}{cccc}
1 &amp; 0 &amp; 0 &amp; 0       \\
0 &amp; 0 &amp; 0 &amp; 0       \\
0 &amp; 0 &amp; 1 &amp; -1       \\
0 &amp; 0 &amp; 0 &amp; 0
\end{array}\right]\cdot\left[\begin{array}{cccc}
d_k \\
0 \\
a_k^{obs} \\
0 
\end{array}\right]
$$
That doesn't make sense because a) the output matrix is too big and b) $a_k^{obs}$ does not form part of the state, $a_k^{true}$ does. It should look more like:
$$
z_k=\left[\begin{array}{cccc}
1 &amp; 0 &amp; 0 &amp; 0       \\
0 &amp; 0 &amp; 1 &amp; +1       \\
\end{array}\right]\cdot\left[\begin{array}{cccc}
d_k \\
v_k \\
a_k^{true} \\
\Omega 
\end{array}\right] = \left[\begin{array}{cccc}
d_k\\
a_k^{true} + \Omega
\end{array}\right]
$$
Notice the change in sign to get the expression for the acceleration measurement to come out correctly.

Q2: What noise is driving your system? Is it just the acceleration?

Your state update equation should look more like:

$$
\hat{x}_{k+1}=\left[\begin{array}{cccc}
1 &amp; dt &amp; dt^2/2 &amp; 0       \\
0 &amp; 1 &amp; dt &amp; 0       \\
0 &amp; 0 &amp; 1 &amp; 0       \\
0 &amp; 0 &amp; 0 &amp; 1
\end{array}\right]\cdot\left[\begin{array}{cccc}
d_k \\
v_k \\
a_k^{true} \\
\Omega 
\end{array}\right]
+ 
Gw_k
$$
where $G$ determines which part of the state is driven by the noise process $w_k$.  The dimension of $w_k$ (is it a scalar or a vector?) will determine whether, if more than one part of the state is being driven, whether those drivers are different for each part.

If the driver is just acceleration, then choose:
$$
G = \left[\begin{array}{cccc}
0 \\
0 \\
1 \\
0 
\end{array}\right]
$$
assuming that the offset in acceleration is not changing.

Then the question of what the covariance of $Q$ is boils down to: how quickly does your acceleration change?  That will really depend on your precise system. In this case, $w_k$ is just a scalar.

As to the covariance of $R$, that will depend on how different your measurements are form reality. I'd need more information to make a stab at that.