The light has teken 13 billion years to reach us, and because of the current curvature of space-time, that translates to particular distance in miles/kilometers/what-have-you. And the “increasing amount of reality”, or, to switch to the proper term, “expanding space-time” between the objects means that, yes, in the future, the distance between them as measured in miles/kilometers/what-have-you will indeed increase to the equivalent 14-, 15-, 16-, billion light-years.
The fallacy of the OP is to extrapolate backwards and say that since the Universe is 13 billion years old and expanding, then 13 billion years ago, the farthest quasars were much closer and the light could not have taken this long to get here.
HOWEVER, general relativity reminds us that the more densely you pack matter and energy, the more you curve the space-time around it. When all the objects in the universe were still quite close to each other, distance and time simply could not be interepreted the way they are now. You simply could not speak of the distance traveled by light over a certain period of time in the way that makes sense to us.
It’s not a problem talking about the future, because the effects of future rarefaction of matter and energy in the Universe will not result in a significant difference in the curvature of space-time from what it is now. But the distant past is an entirely different story. We see the distant quasar the way it was 13 billion years ago because that is how long the light it emitted back then has taken to reach us. Because of the current curvature of space-time and the speed of light within that current curvature, we can translate that into a measurable distance.
The middle part of your post is pretty much correct, but doesn’t really deal with what I’m addressing above.
I think the detailed discussion of “dark matter” in this thread is somewhat tangential, as it originates from a confusion by Roland Deschain between it and “dark energy”, which RyanD004 has already pointed out.
I’ll just expand on that by mentioning that some of the original mathematical results obtained by Einstein and his followers suggested that empty space would inherently possess a repulsive gravitational force. One might assume that further refinement of the theory would iron out such an obviously nonsensical result.
However, recent measurements have suggested that objects distant from each other may indeed repel each other gravitationally, owing to something previously unseen. This “dark energy” may vindicate the “negative vacuum gravity” (my phrase) that originally popped up, since cosmological expansion puts more empty space between objects, which would increase the push.