First off, if Earth were point B, and you were an observer at point A looking at it with the most magnificent telescope ever imagined, you would still not see the Earth, because it didn't exist 13 billion years ago. I assume you picked 13 billion years because it is roughly the age of the universe, so you'd see the universe as it existed then, but that doesn't get you what I think you want. I will instead pick 4.5 billion years ago (or objects A and B 4.5 billion light years apart), because then you will have an actual Sun and Earth to look at but long before life evolved.
We'll also wave our arms over the tracking software of your telescope being able to locate Sun and Earth in their orbits over the course of 4.5 billion years as seen from point B.
Yes. You would have to wait 4.5 billion years to see the observer at point A looking back at you through his telescope. You would now be looking at a cooling hunk of molten rock with no moon or oceans.
No. If you waited 4.5 billion years and saw Observer A, he would have been dead for 4.5 billion years by the time you see him. In fact, his Sun would likely be expanding to the point of burning his planet to a cinder if not engulfing it entirely. But of course, you would have to wait another 4.5 billion years to see that. You would be observing the photons that bounced off Observer A all that time ago and have been traveling ever since. It is important to note that this is not him, and does not mean he still exists.
Hopefully this answers your question, though it doesn't have much to do with Special Relativity, it's still good blow-your-mind stuff. It's just speed of light stuff which we really knew and measured (albeit inaccurately) long before Einstein.
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