I am having trouble in understanding Theorem 8.2 of Adams's book, regarding the spectral sequence of a cohomology theory of a spectrum $E$ where $X$ is an infinite dimensional CW-complex. I think I should prove somehow that the spectral sequence in this case will be

$$\varprojlim_{a}{}^{p}E^{q}(X_{a})\implies E^{p+q}(X)$$ where $X_{a}$ are the skeletons of $X$, $\varprojlim_{a}{}^{p}$ is the $p-$th right derived limit as defined in the same chapter of Adams. As far as I understood, we start with a spectral sequence with the first page being $E_{1}:=E^{p+q}(X_{p},X_{p-1})$. So, I have many questions.

1. How do we prove that the second page is given by $\varprojlim_{a}{}^{p}E^{q}(X_{a})$?
2. What is the notion of "convergence" here? Is it the condition one he mentions which says $E_{\infty}^{p,q}\to\varprojlim_{r}E_{r}^{p,q}$ is isomorphism? And how are the maps $E_{r+1}^{p,q}\to E_{r}^{p,q}$ defined and why being monomorphism implies that the limit exists?
3. What are exactly the filtration quotients of $E^{p+q}(X)$ in condition 3? And how is the exact sequence constructed (the exact sequence in condition 3)?
4. How do we even use theorem 8.2 in Adam's book to provide the exact sequence? This means, how do we verify condition (ii) of this theorem?

I studied the Atiyah-Hirzebruch spectral sequence in Kochman's book which is beautifully presented, but the infinity case only appears in the sketchy book of Adams unfortunately.