Question 1. You mean $\hat{x}[0]$ and $P[0]$? They are just guesses. You can use your first measurement for $\hat{x}[0]$ and give a possible error value $P[0]$ depending on your measurement accuracy. $k$ starts from 1.
Question 2. You use the $i$th column of the matrix.
Question 3. You have to have an $h$, maybe it is just the identity function and you measure all the states? $u_m[k]$ is just the input, they just give a different name for unknown reasons I think.
Question 4. These choice of weights are independent from the index.
Question 5. $R[k]$ is the covariance of the measurement error. You can use the accuracy knowledge of the measurement device to determine it.
Question 7. $k$ is just a typo. You can use something like the LU decomposition to solve for $K$. But generally the number of outputs are small, so $P_y$ is a small square matrix.
Question 9. $u_s[k]$ is just the input.
Question 11. $Q[k]$ is the process noise covariance, which is something you know as in the linear KF.
I believe Wikipedia has a good explanation: https://en.wikipedia.org/wiki/Kalman_filter#Unscented_Kalman_filter
A short stability proof of the Kalman filter is given in section 2.4.1 of:
Rhudy, M. B. (2013). Sensitivity and stability analysis of nonlinear
Kalman filters with application to aircraft attitude estimation. West
Virginia University.
I will give a short summary of what is stated in this dissertation. First of all it is assumed that $Q$ and $R$ (these two matrices can also be time varying) are such that $(F_k,Q)$ is uniformly controllable and $(F_k,H_k^\top R^{-1} H_k)$ is uniformly observable, then the error covariance matrix $P_k$ has a finite positive definite upper and lower bound.
The a priori estimation error $\tilde{x}_k = \hat{x}_k - x_k$ can shown to be stable using the following Lyapunov function
$$
V(\tilde{x}_k) = \tilde{x}_k^\top P_k^{-1} \tilde{x}_k,
$$
which is guaranteed to be a positive definite function when the controllability and observability conditions are satisfied. The estimation error dynamics is given by
$$
\tilde{x}_{k+1} = F_k\,(I - K_k\,H_k)\,\tilde{x}_{k} + F_k\,K_k\,v_k - w_k.
$$
For the bounded input bounded output (BIBO) stability analysis it is only required to consider the homogenous part of the error dynamics. Doing so yields to following increment of the proposed Lyapunov function
\begin{align}
\Delta V(\tilde{x}_k) &= \tilde{x}_{k+1}^\top P_{k+1}^{-1} \tilde{x}_{k+1} - \tilde{x}_k^\top P_k^{-1} \tilde{x}_k, \\
&= \tilde{x}_k^\top \left[(I - K_k\,H_k)^\top F_k^\top P_{k+1}^{-1} F_k (I - K_k\,H_k) - P_k^{-1}\right] \tilde{x}_k,
\end{align}
which after some more manipulation can shown to be negative definite. Previously it was shown that $P_k$ is upper and lower bounded and therefore $P_k^{-1}$ should be as well, which together with $\Delta V(\tilde{x}_k)\prec 0$ enables one to show that the Kalman filter is BIBO stable.
For showing that the Kalman filter in stable one only needs that $w_k$ and $v_k$ are bounded, so nowhere is it required that $Q$ and and $R$ are such that $w_k \sim \mathcal{N}(0, Q)$ and $v_k \sim \mathcal{N}(0, R)$. So the stochastic terms can have very different covariance matrices associated with them. Since the Kalman filter is dealing with a stochastic system an more in-depth analysis is needed to show that covariance of the error remains bounded when there is a mismatch between noise covariance matrices the actual noise compared to the ones used in the Kalman filter. Further resources regarding this are mentioned earlier mentioned dissertation.
However, having a mismatch in the noise covariance matrices can result in a far from optimal noise suppression, as is discussed in:
Ge, Q., Shao, T., Duan, Z., & Wen, C. (2016). Performance analysis of
the Kalman filter with mismatched noise covariances. IEEE Transactions
on Automatic Control, 61(12), 4014-4019.
Best Answer
Assuming that your model of the dynamics is correct (i.e. $F_k$, $B_k$ and $H_k$) then the good news is that any (positive semi-definite) choice for $Q$ and $R$ will result in a state estimator that is stable, so the estimated state converges to the true state in the absence of disturbances $w_k$ and $v_k$ and converge to some bound around the true state when there are disturbances.
If the covariance matrices are off by some scalar factor $\alpha$, then one still yields the same state estimate. This can be seen by looking at the matrix update equations from the Kalman filter
\begin{aligned} \mathbf {P} _{k\mid k-1}&=\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k},\\ \mathbf {S} _{k}&=\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k},\\ \mathbf {K} _{k}&=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1},\\ \mathbf {P} _{k|k}&=\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}. \end{aligned}
Namely, instead of $\mathbf {P} _{k|i}$, $\mathbf {Q} _{k}$ and $\mathbf {R} _{k}$ using $\mathbf {P'} _{k|i}=\alpha\mathbf {P} _{k|i}$, $\mathbf {Q'} _{k}=\alpha\mathbf {Q} _{k}$ and $\mathbf {R'} _{k}=\alpha\mathbf {R} _{k}$ respectively yields
\begin{aligned} \mathbf {P'} _{k\mid k-1}&=\mathbf {F} _{k}\mathbf {P'} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q'} _{k} = \alpha\left(\mathbf {F} _{k}\mathbf {P} _{k-1\mid k-1}\mathbf {F} _{k}^{\textsf {T}}+\mathbf {Q} _{k}\right),\\ \mathbf {S'} _{k}&=\mathbf {H} _{k}\mathbf {P'} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R'} _{k} = \alpha\left(\mathbf {H} _{k}\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}+\mathbf {R} _{k}\right),\\ \mathbf {K'} _{k}&=\mathbf {P'} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S'} _{k}^{-1} = \frac\alpha\alpha\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\textsf {T}}\mathbf {S} _{k}^{-1} = \mathbf {K} _{k},\\ \mathbf {P'} _{k|k}&=\left(\mathbf {I} -\mathbf {K'} _{k}\mathbf {H} _{k}\right)\mathbf {P'} _{k|k-1} = \alpha\left(\mathbf {I} -\mathbf {K} _{k}\mathbf {H} _{k}\right)\mathbf {P} _{k|k-1}. \end{aligned}
So the gain matrix $\mathbf {K'} _{k}$ remains the same under any scalar factor $\alpha$, so the state update step will also remain the same. One might run into some numerical issues when $\alpha$ is really big or really small, but on the other side $\alpha$ can also be used for normalization and make the Kalman filter more numerically robust. Such normalization will also change the direct relationship between $\mathbf {P'} _{k|k}$ and the uncertainty of the state estimate, so keep the factor $\alpha$ in mind when interpreting $\mathbf {P'} _{k|k}$.
I can not answer your more general question of the exact impact of using covariance matrices that deviate is more ways that just a scalar factor. It can be noted that chosing the $Q$ and $R$ matrices is often also considered a tuning step of making a Kalman filter. These matrices can also be used to reduce the impact of other types of disturbances (so other than zero mean Gaussian white noise).