Wrong. There could be tons of different things going on inside, absolutely indistinguishable from outside, which only sees mass, electric charge and angular momentum.

Also entropy. Anyway, those are determined by the mass, electrical charge and angular momentum of the matter that fell inside. We may not want to call it a causal connection, but it's certainly a case of properties within determining properties outside.

There is no causal connection from inside to outside whatsoever, barring FTL communication.

There is no direct causal connection, meaning a worldline from the inside to the outside of the black hole. But even if the horizon screens almost all of the infalling matter properties, it doesn't screen everything (and probably, but this is a matter of quantum gravity, doesn't screen nothing).

Wrong again. There is no singularity of any kind behind the cosmological horizon (which is not a closed surface to begin with, so it cannot "surround" anything). Well, there might be black holes and stuff, or there might not be, but there is certainly not a requirement of anything singular being there. Consider googling the definition of singularity in general relativity.

I'll admit to not have much knowledge about this specific theme, and I'll educate myself more properly, but in the case of my earlier sentence I used "singularity" as a mathematical term, referring to a region of spacetime in which the GR equations acquire a singular value, so not specifically to a gravitational singularity like a black-hole or a domain wall. In the case of most commonplace cosmological horizons, this region is simply space-like infinity.