It is true that from the perspective of an observer inside the event horizon of a black hole, the photons falling into the black hole would appear to be infinitely blueshifted and would arrive at the singularity at the center of the black hole all at the same time. This is a consequence of the extreme gravitational forces present in the region, which cause the photons to be blueshifted and to follow highly curved paths as they fall towards the singularity.
However, from the perspective of an observer outside the event horizon, the photons falling into the black hole would not be infinitely blueshifted, and would not all arrive at the same time. From this perspective, the photons would follow paths that are determined by the curvature of spacetime, which would generally not be straight lines.

No, "loss of causality information" and "infinitely blueshifted" is not the same thing.
The event horizon of a black hole acts as a "one-way membrane" that separates the region inside the black hole (the interior) from the region outside (the exterior). Once an object or particle crosses the event horizon, it becomes trapped inside the black hole and cannot escape. This means that, from the perspective of an observer outside the black hole, the interior of the black hole is causally disconnected from the exterior. This means that there is no way for an observer outside the black hole to receive information about what is happening inside the black hole, or to affect events that are occurring inside the black hole. This is what is meant by "loss of causality information."
The phenomenon of an object being infinitely blueshifted as it approaches the event horizon of a black hole is a consequence of the extreme gravitational forces present in the region. more extreme. In this sense, it is possible to say that the object is "infinitely blueshifted" as it approaches the singularity. However, this is not the same thing as a loss of causality information, as the object is not able to transmit any information about its experience to an outside observer once it crosses the event horizon.

The choice of coordinates used to describe a physical system is a matter of convenience, and different choices of coordinates can give rise to different perspectives on the same physical situation. It is generally more useful to choose coordinates that are well-behaved (e.g. that do not diverge or become singular at certain points) and that are adapted to the symmetries of the system (e.g. cylindrical coordinates for a cylindrically symmetric system).

With general relativity, the gravitational field is encoded in the curvature of spacetime. This means that the path that light follows is determined by the curvature of spacetime, which is itself determined by the distribution of mass and energy in the universe.
So with a black hole, the spacetime around the black hole is highly curved due to the presence of the black hole's mass. This means that light that is emitted by one observer will follow a path that is determined by the curvature of spacetime, which will generally not be a straight line.
It is possible to consider a "1+3" decomposition of spacetime, where space is divided into a set of spatial coordinates and time is treated as a separate coordinate. However, this decomposition is not unique, and different choices of coordinates can give rise to different perspectives on the same physical situation.
With a black hole, it is possible to consider a set of coordinates that are "equidistant" from the center of the black hole, as you described. However, these coordinates would not necessarily be a good choice for describing the physics of the system, as they would not be well-behaved at the event horizon of the black hole.
In general, the physics of a system should not depend on the choice of coordinates used to describe it. Therefore, it is generally not valid to draw conclusions about the behavior of a physical system based on a particular choice of coordinates.