I see. Then, if you simplify the setup as you do, of course no interference patterns occur (neither on Screen 0 nor on Screen 1). That's, because the photons that interfere on Screen 0 and Screen 1 are from uncorrelated sources since the preparation of entangled photon pairs through parametric down conversion is an entirely spontaneous, i.e., completely random process. So the two entangled pairs used in the experiment are completely uncorrelated with each other, i.e., you superimpose two completely incoherent light sources at Screen 0.
That's also confirmed, BTW, by the original setup: If you take all 4 subsensembles together, i.e., add the 4 plots for ##R_{0j}##, which represents all photons detected with ##D_0##, you'll find no two-slit interference pattern.
The intriguing thing with the entangled photons is that you can by choosing the 4 subensembles, decide whether you look at situations, where you know from which of the two slits the original photons came. E.g., if you take the ensemble, where one of the "blue" photons which is registered by ##D_3##, then you know for sure this photon was parametric-downconverted from a photon which came from the lower slit ("which way information"). This in turn ensures that the photon state for the photon registered at ##D_0## is due to an incoherent mixing of two independent photon sources, and thus there is no double-slit interference pattern (plot for ##R_{03}##). Interestingly, however, theres the single-slit interference pattern (i.e., the sine-like curve shown in this plot). That's because the photons being produced from the photon coming through the upper slit are a coherent source for themselves and thus show the single-slit interference pattern and so are the photons being down-converted from the photons coming through the lower slit, showing the same single-slit interference pattern. Their incoherent addition thus shows the single-slit interference pattern.
The same holds true if you use the subsensemble where one of the "red photons" was registered by detector ##D_4##, because then this photon for sure originated from one of the entangled photons created by parametric downconversion of a photon that came through the upper slit. Correspondingly, you know that the photon state for the photon registered at ##D_0## for this subensemble is described by an incoherent mixing of two independent photon sources and thus there's no interference pattern (or rather only the single-slit interference pattern).
If you look at the ensembles prepared by looking only at photons where ##D_1## registered a photon, the situation is very different: Now you don't know, whether this photon orinated from one of the photon pairs which was created by the upper or the lower slit, i.e., for this ensemble there's no which-way information. Due to the entanglement of the pairs (12) and (34) now the photon state at detector ##D_0## is a coherent superposition of indistinguishable photons, and thus an interference pattern is observed (see the plot for ##R_{01}##). The same argumentation holds for the ensemble, where one of the photons has been registered with detector ##D_2##. You have again an interference pattern, as seen in ##R_{02}##, but it's anticorrelated to the one of ##R_{01}##, i.e., where you have constructive interference in ##R_{01}## you have destructive interference in ##R_{02}## and vice versa. Adding up ##R_{01}## and ##R_{02}## thus doesn't show any interference pattern (or, as shown in the original paper, the single-slit interference pattern).
Now all the manipulations of the equipment you took out of the picture do not causally influence the photons registered at ##D_0##, and thus you can expect that in your experiment you don't see a double-slit interference pattern. That is, however, also explained independently as written in the first paragraph.
The paper by Kim et al is really very nicely readable:
https://arxiv.org/abs/quant-ph/9903047
Also ref. [14] therein is a very detailed explanation of the theory of parametric downconversion:
https://doi.org/10.1103/PhysRevA.50.5122
