It's independent of mass because it's just a little extra space being created everywhere all the time. It's not exactly a force. First, there was 1 meter between them. Now there's 1 meter and a tiny bit more. The amount of space being created is, in theory, the same everywhere. It's not a repulsive force- space is just being created.

It can actually end up making things very far away appear to move faster than light away from us, since there's a lot of space in between.

Yes, any two bound objects will do so, irrespective of their mass or anything else.