Suspecting that the universe is even more bigger

Yes. It is completely plausible that the universe is infinite in spatial extent, despite the fact that any observer can observe only a finite part of it. In fact, current standard cosmological models generally assume that it is infinitely large, because observations are consistent with zero spatial curvature. (There is nonzero spacetime curvature.) You can read about a $k=0$ Friedmann universe.

If the universe actually has positive spatial curvature, then it has finite size. But the spatial curvature has to be so small to match observations that the whole universe would be much larger than the part we can see.

No, it is not plausible that the extraterrestrial scientists would see such differences when they look in opposite directions.

The Friedmann universes describing our expanding space are derived under the assumption that the universe is homogeneous and isotropic. In other words, it is the same wherever you are and in whichever direction you look. In this respect, it doesn't matter if the universe is infinite or finite.

Because you mention galaxies 13 billion light years apart, and that happens to be the approximate age of the universe, I think you probably believe that is the size of the observable universe. However, due to the universe's expansion, that is not the case. It's more like 42 billion light years in radius. (See the answer to this question for a good explanation: Size of the universe)

I'm also assuming that you are asking what someone at the edge of our observable universe would see, not realizing that we are also at the edge of their observable universe. Again, the answer is that observers anywhere in the universe would all see the same sorts of things (ignoring small-scale variations, of course).