From the Earth If we observe two galaxies that are diametrically opposed and each 1000ly far from the earth. the separation distance between the galaxies will be 20000ly? really question is: if the galaxies were separated from Earth 10Gly. then the separation distance between galaxies would 20Gyr?
For nearby distances, yes, you can just add them up in the usual way. (Of course, if two galaxies are just a few thousand light-years apart, then they're right on top of each other -- galaxies are much bigger than 1000 light-years!)
For large distances (i.e., distances comparable to the Hubble length, as in your last example), you have to be careful. The best answer is that there is no unique, well-defined notion of distance over such large distances. On cosmological distances, spacetime is curved, and what that means is that there are no inertial reference frames covering these large distances. Different people may choose different (non-inertial) reference frames, and as a result they'll disagree about the distance, but no one is necessarily "right."
When people talk about the distance to a faraway galaxy, they most often mean distance as measured in a particular coordinate system, namely comoving coordinates, with distances evaluated at the present (cosmic) time. With that specific definition, the answer to your question is yes: two galaxies that are 10 Gly from earth in opposite directions are 10 Gly from each other.