I don’t want to be pedantic or anything just make communication easier:
The moving cursor is called “transform gizmo”.
It is displayed where the “pivot point” is.
You can set the pivot point to be at different places. You do that in one of the dropdowns in the bar above the viewport. Just hover your mouse above all the drop downs. They will display a tooltip. The dropdown whose tooltip says “Transform Pivot Point” is the one where you set it.
In your case it is probably set to “median point”.
Now if that is case the pivot will be where the objects origin is. The origin is the little orange dot which each object has. You appear to have turned it invisible. You can turn it on in the overlays dropdown, which is also in the bar above the viewport.
The reason that after exporting and importing the origin has shifted is because the obj exporter appears to reset it to 0/0/0 in world coordinates. I don’t use obj so I am not sure how to resolve this besides using a different forma. Obj is antiquated. USD or FBX is usually a better choice.
However, you should probably get familiar with the concepts of pivot points and origins:
And on the other hand… you don’t need to export a object as OBJ to transfer from one blender file to another… you can File → Append it directly from a blend file…
Ironically, now after trying again now that you mentioned it, it worked. May be the memory was so full that I couldnt even copy to the clipboard or something
I’m not sure copy/paste of objects involves memory. I think it moves a duplicate of the object to another .blend file on your system somewhere. Then when you quit blender completely, open it again, and press ctrl+v it appends the object from that special .blend file.