Derivation of the formula for longitude of ascending node for a satellite
I think you're not too far from understanding the formula. However, there are a couple details you're missing: 


From what I understand from the document you linked to, $t_0 = t_{oe}$ is the time at the beginning of the week, or reference epoch.
When computing an angle from a angular velocity, you usually need to compute it with respect to an initial angle and an initial time: when the Earth and the satellite rotate between $t_0$ and $t$, the angle they rotated of is $\dot{\Omega}(t - t_0)$ or $\dot{\Omega}_E(t - t_0)$.
The only exception to this rule seems to be the initial Greenwich longitude with respect to the vernal equinox, which seems to be computed from $t = 0$.
$w_0$, the initial Greenwich longitude wrt the vernal equinox, is $\dot{\Omega}_E t_{oe}$. From this formula, it seems like the document considers an origin of time $t = 0$ when the longitude of Greenwich wrt to the vernal equinox at this time is 0.


To summarize: the formula would have been more explicit and comprehensible written as:
$$
\Omega(t) = \Omega_0 - \dot{\Omega}_E t_{oe} + (\dot{\Omega} - \dot{\Omega}_E)(t - t_{oe})
$$
with:  


$\Omega_0$ = LAN wrt the vernal equinox at $t_{oe}$
$\dot{\Omega_E}$ = angular velocity of the Earth wrt the vernal
equinox
$-\dot{\Omega}_Et_{oe}$ = Longitude of Greenwich at $t_{oe}$
$\dot{\Omega}$ = angular velocity of the satellite wrt the vernal
equinox
$(\dot{\Omega} - \dot{\Omega}_E)(t - t_{oe})$ = relative angular
displacement between the satellite and Greenwich, between time $t$ and
$t_{oe}$


That said, it seems like really bad practice to name $\Omega(t)$ the longitude with respect to Greenwich and $\dot{\Omega}$ the angular velocity with respect to the vernal equinox. This causes quite a bit of confusion, since mathematically speaking, for any linear function $f$, $f(t) = f(t_0) + \dot{f}\times(t - t_0)$