Special relativity theory says simultaneity is relative, meaning that different observers will not agree on what happened first and what second. Does it then make sense to say that looking at distant stars, we see them how they looked "billions of years ago" and not how they look now? Does it make sense to talk about what these stars look like now? How do we define this "now" if simultaneity is relative?
Consider two points A and B in spacetime. If they are separated by a spacelike interval, then indeed different observers can disagree on which happened earlier. However, there are two other possibilities. A could be inside the future light cone of B, in which case all observers agree that A is in B's future. Similarly, B can unambiguously be in A's future.
Things may get more complicated in general relativity, but this idea still holds. The distant galaxies we see are in our past, and no shift of reference frame can alter this. Causality is preserved.
Yes, in the framework of Special Relativity, it makes perfect sense that there is a present state for all the stars that we see. There is an enormous set of events (in the ordinary sense) happening at this precise moment all around the Universe. But...
But for another observer moving with respect to us, the set of things that are happening at the same time IS different. I remark the word "IS", not to be confused with "looks". There is a present state of the world for another observer that moves relative to us, and is a state as real as ours, but just different.
This is only valid in Special Relativity. As Alfred Centaury correctly warns in its comment, the definition of "the present" is a convention in FLRW Cosmology (within the framework of General Relativity). The Universal Time of the Comoving observers is choosen because it seems rather natural (all observers could for example set his watches to a certain value when they measure a given mean mass density in their local volume) but spacetime could be sliced in another way, defining a different set of synchronized observers.
More formally, the definition of time in cosmology is motivated by the Robertson-Walker metrics, because fixing any particular time value leaves a spatial metric that is homogeneous and isotropic.