Understanding the $\phi^4$ phase diagram 
I'm having trouble making sense of this phase diagram. The model is a $V(\phi)=g_2 \phi^2+g_4\phi^4$ scalar field theory. Here's what I think I understand: the capital letters represent different phases (the B and D phases have the hatched areas associated with them, while all other phases occupy only lines or points in the phase diagram rather than entire areas). G is the Gaussian fixed point and WF the Wilson-Fisher fixed point. Define $\ell=\log\left (\frac{\Lambda_0}{\Lambda}\right)$, where $\Lambda_0$ is the original cutoff of the theory and $\Lambda$ is the scale that you're viewing it (you've integrated out all the modes between $\Lambda_0$ and $\Lambda$, $\Lambda<\Lambda_0$). The arrows in the trajectories on the phase diagram point towards increasing $\ell$.
The first question is did I get the above description correct?
Second, to take the continuum limit $\Lambda_0 \rightarrow \infty$ but keep the same scale you're viewing the system $\Lambda$, $\ell$ must then go to $\infty$. That suggests to take the continuum limit from any point on one of the trajectories, you just follow the arrows. But that seems to suggest that only systems on the line going through the points G and WF have continuum limits, since everywhere else the arrows are leading to infinite mass $g_2$ which seems to be nonsense. However, I have my doubts about saying that only the line G-WF has a continuum limit, because the author (Hollowood) makes the statement that the region to the right of the line joining C and E has no continuum limit, implying that everything to the left of that line does have a continuum limit. 
Third, consider changing the scale that you look at $\Lambda$. Increasing this scale (probing the UV) while keeping $\Lambda_0$ fixed amounts to decreasing $\ell$, so you would follow the arrows backwards to see what happens in the ultraviolet. In regions B and D the UV behavior is sensible (finite values of mass and coupling - zero in fact), but the IR behavior (in the direction of the arrows) is nonsensical as parameters are flying off to infinite. If you have one theory that is supposed to describe everything at all scales, then do all trajectories must begin and end at finite fixed points?
Lastly, it seems increasing $\Lambda$ and keeping $\Lambda_0$ fixed, or keeping $\Lambda$ fixed and decreasing $\Lambda_0$, both decrease $\ell$. Does this mean keeping the grid the same but viewing it at a increased resolution is indistinguishable from putting the system on a coarser grid but keeping the resolution the same?
 A: 1) About description of different phases consult end of chapter 2 of 6 Lectures on QFT, RG and SUSY, here I reproduce this description: 
Continuum theories are limited to the grey region.
Also useful to see chapter about renormalization group from David Skinner: Quantum Field Theory II
2) Theories in white region of this diagram have not continuum limit:
All couplings in any theory in white region diverge as we try to follow the RG back to the UV; these theories do not have well–defined continuum limits.
3) If in IR parameters like mass go to infinity, this mean that you have trivial theory in deep IR, without any propogating excitations (in our example. In general you can end with Namby- Goldstone modes or non-trivial TFT). If you wanna to have some interesting theory in deep IR, this means that you must have massless particles, because only such excitations will play role in such limit. Free theories usual are scale-invariant, and symmetry usually extend to CFT (counter-example is Maxwell theory in 3d). 
So in deep IR if you wanna have non-trivial theory, you must finish at some IR fixed point, $CFT_{IR}$.
For more complete and general discussion of IR physics see video lectures by Zohar (around 6:00).
4) I agree with your interpretation.
