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Yes, you are correct on the first one. The second one would depend on a few different variables, but in general if you are correct on that one too.I have heard that the higher elevation you are the faster in time you are traveling than someone lower in elevation. Immeasurable on Earth but its the same theory suggesting if you travel to another galaxy then return to Earth like 1 year later everyone you know would be old or dead.
The simple way of explaining it (how I understood it) was that light has to travel a greater distance near objects that displace/alter spacetime. I think of it sort of like a bubble in space, but the exact opposite. You have all these atoms smashed into a given area. That mass (earth) affects the curvature of spacetime. The more curve, the greater the distance the light has to travel. Since we are on a big chunk of atoms called Earth, it messes up the curvature of spacetime. I always get mixed up on which is which, but I just remembered that in empty space (thing a big chunk of nothing and darkness), light moves in straight lines and is not altered so it will move faster.
It's sort of like which is a shorter distance:
-Walking over the surface of the earth to Africa
Or
-Walking in a tunnel through earth into the ground at Africa.
Walking into the tunnel in the ground would get you there faster, and to the same location. You could walk at the same speed over Earth to get to Africa, buy it would take longer.
But another crazy thing is that even if you got to the same village/spot in Africa, and you could theoretically do both experiments, you would only be at the same location on Earth. Keep in mind the universe is expanding, and our planet is orbit a star, and our solar system is in a spiral galaxy which is also rotating. Every day you wake up on a different location in the universe, you just so happen to be on the same bed!
Here is a good spacetime GIF: https://pin.it/5F3amYRjw