Start with the Taylor expansion
$$
u(x,t+k)=u(x,t)+u_t(x,t)k+\frac12u_{tt}(x,t)k^2+...
$$
Use the PDE to convert time into space derivatives, $u_t=-[f(u)]_x$, then
\begin{align}
u_{tt}&=-[f(u)]_{xt}=-[f'(u)u_t]_x\\
&=[f'(u)[f(u)]_x]_x\\
\end{align}
Now realize this derivative structure using symmetric divided differences and a half-step scheme. Let the index denote the offset in $x$ direction in units of $h$
$$
[f'(u)[f(u)]_x]_x=\frac{f'(u_{+1/2})[f(u_{+1/2})]_x-f'(u_{-1/2})[f(u_{-1/2})]_x}h
$$
The $x$ derivatives at the half-steps are again approximated using central divided differences with step size $h/2$,
$$
[f'(u)[f(u)]_x]_x=\frac{f'(u_{+1/2})\frac{f(u_{+1})-f(u)}h-f'(u_{-1/2})\frac{f(u)-f(u_{-1})}h}h
$$
and finally the midpoint arguments of the derivative of $f$ by $u$ can be approximated by the mean of the neighboring integer grid values, $u_{\pm 1/2}=\frac12(u+u_{\pm 1})$.
If one inserts this in backwards, using the full-step central derivative for the degree-1 term, $[f(u)]_x=\frac{f(u_{+1})-f(u_{-1})}{2h}$ one gets back to the given formula of the method.
The proposed values of $\chi$ in a) are correct. One notes that the method is a straightforward finite-difference approximation of the advection-diffusion method $u_t + au_x = \chi u_{xx}$. Introducing the Courant number $r = |a| {\Delta t}/{\Delta x}$, the diffusion constants from a) rewrite as
$$
\chi \in |a|\frac{\Delta x}{2} \times \left\lbrace \frac{1}{r},\,
1,\,
r,\,
0 \right\rbrace ,
$$
which correspond to the numerical viscosities of the Lax-Friedrichs, upwind, Lax-Wendroff, and central explicit method, respectively. One notes that if $0<r<1$, then those values of $\chi$ are sorted in decreasing order.
As done in this post, let us assume a perturbation of the form $u_j^n = \xi^n e^{-\text i k j \Delta x}$, where $\xi = \text e^{\text i \omega \Delta t}$. Note that $\omega$ is a complex number, with real part $\omega_R$ and imaginary part $\omega_I$. Injecting this Ansatz in the time-stepping formula, Euler's formulas lead to
$$
\xi = 1 + \text i r \sin( k \Delta x) + 2\lambda \left( \cos(k \Delta x) -1 \right) ,
$$
where $r$ is the Courant number and
$$
\lambda = {\chi}\frac{\Delta t}{\Delta x^2} \in \left\lbrace \frac{1}{2},\,
\frac{r}{2},\,
\frac{r^2}{2},\,
0 \right\rbrace
$$
denotes the Fourier number. From this equation, one equates the real and imaginary parts:
\begin{aligned}
\text{Re}\, \xi &= e^{-\omega_I \Delta t}\cos(\omega_R \Delta t) = 1 + 2\lambda \left( \cos(k \Delta x) -1 \right) ,\\
\text{Im}\, \xi &= e^{-\omega_I \Delta t}\sin(\omega_R \Delta t) = r \sin( k \Delta x) \, .
\end{aligned}
The stability can be analyzed in terms of the squared amplification factor $|\xi|^2$ which has to be smaller than one. Hence, we must have
$$
|\xi|^2 - 1 = 4\lambda^2 - 4 \lambda + r^2 + 4 \lambda (1-2\lambda) Z + (4\lambda^2 - r^2) Z^2 \leq 0
$$
for all $Z = \cos( k \Delta x)$ in $[-1, 1]$.
The dispersion is analyzed in terms of the phase velocity ${\omega_R}/{k}$ with
${\omega_R}\Delta t = \arctan\left[{\text{Im}\, \xi}/{\text{Re}\, \xi}\right]$.
The attenuation is analyzed in terms of
${\omega_I}\Delta t = -\ln |\xi|$
which is non-negative if $|\xi| \leq 1$ (cf. e.g. (1) p. 182).
(1) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996. doi:10.1007/978-1-4612-0713-9
Best Answer
For the first part, these slope- or flux-limiter schemes are not second-order accurate anymore. In fact, their order of accuracy measured numerically is rather around 1.6-1.8. This becomes obvious after realising that they are a kind of compromise between first-order and second-order schemes, see e.g. the 2002 book by R.J. LeVeque.
For the second part, I may find some special cases where this property is proven in literature. To be updated.