I believe to some extent there is confusion due to the name of the scheme.
The "classic" Lax-Friedrichs Scheme is model-agnostic in the sense that no information on the flux function $f(u)$ is required.
You present what is sometimes labeled as "Local Lax-Friedrich" or Rusanov Scheme which includes information on the flux function. For systems $\big (\boldsymbol{f} \in \mathbb{R}^m, m >1 \big)$ these are the eigenvalues $\mu^{(p)}, p = 1 \dots m$ of the Jacobian $\frac{\partial \boldsymbol{f}}{\partial\boldsymbol{u}}$.
In one dimension, this boils down to the scalar quantity $f'(u)$.
You can look up the schemes for 1D in the chapter of the chapter I already linked. For a general introduction to the topic, I would point you to the entire lecture notes.
But now regarding your question:
You have the finite volume scheme
$$ \boldsymbol{u}_j^{(n+ 1)} = \boldsymbol{u}_j^{(n)} + \frac{\Delta t}{\Delta x} \Big( \boldsymbol{F}_{j - 1/2}^{(n)} - \boldsymbol{F}_{j+ 1/2}^{(n)} \Big)$$
and want to use the "Local" Lax-Friedrich or Rusanov scheme to compute the numerical fluxes $\boldsymbol{F}_{j\pm 1/2}$.
Your general formulation of the scheme is correct:
$$\boldsymbol{F}_{j+ 1/2}^{(n)}\Big(\boldsymbol{u}_l^{(n)}, \boldsymbol{u}_r^{(n)}\Big) = \frac{\boldsymbol{f}\Big(\boldsymbol{u}_l^{(n)} \Big) + \boldsymbol{f}\Big(\boldsymbol{u}_r^{(n)} \Big)}{2} - \frac{\alpha}{2} \Big( \boldsymbol{u}_r^{(n)} - \boldsymbol{u}_l^{(n)} \Big) $$
where the indices fulfill $l = j, r = j + 1$.
For the "classic" Lax-Friedrichs scheme, $\alpha$ is exactly the same as in the scalar case: $$\alpha_{LF} = \frac{\Delta x}{\Delta t}$$
For the "local" Lax-Friedrichs / Rusanov scheme, I am aware of the following equation for $\alpha$:
$$\alpha_{\text{Rusanov, } j + 1/2} = \max \Bigg \{ \max_p \Big \{ \Big\vert \mu_l^{(n, p)} \Big\vert \Big\}, \max_p \Big \{ \Big\vert \mu_r^{(n, p)} \Big\vert \Big\} \Bigg\} $$
i.e., you just consider the maximum of the largest absolute eigenvalue of the matrices $\frac{\partial \boldsymbol{f}}{\partial\boldsymbol{u}}\Big \vert_{ \boldsymbol{u}_l^{(n)}} \quad \frac{\partial \boldsymbol{f}}{\partial\boldsymbol{u}}\Big \vert_{ \boldsymbol{u}_r^{(n)}}$ with spectra $\boldsymbol{\mu}_l^{(n)} = \sigma \Big ( \frac{\partial \boldsymbol{f}}{\partial\boldsymbol{u}}\Big \vert_{ \boldsymbol{u}_l^{(n)}} \Big), \boldsymbol{\mu}_r^{(n)} = \sigma \Big ( \frac{\partial \boldsymbol{f}}{\partial\boldsymbol{u}}\Big \vert_{ \boldsymbol{u}_r^{(n)}} \Big) $.
In particular, you are not considering the entire domain, but only the spectra of the Jacobians evaluated at the values at both sides of the interface.
Coming to your second question regarding the CFL condition:
Here a maximization over all interfaces (where you compute the numerical fluxes) is indeed required:
$$ \Delta t \overset{!}{<} \text{CFL} \frac{\Delta x}{\max_j \big \{ \alpha_{\text{Rusanov, } j + 1/2} \big \}} $$
Where the $\text{CFL}$ number is problem dependent.
