Funny isn't it, yes, we live at the seaside but we live on the coast, also can say by the seaside and by the coast, or by the water's edge and at the water's edge, not sure why these differences occur so some may be regional? BY means alongside, AT suggests a point so maybe AT the coast suggests a specificity which clashes with the visualisation? You can say 'a house along the coast' but not 'along the seaside'.
True about 'at' being more specific... eg. I live at the northern most point on the coast
Which would not work as on, by, or anything else
Another accepted translation: 'She lives IN the west coast'. No idea how to imagine anyone living in the coast (is that in the basement of a beach house?
Yes, it sounds strange to me too, but I would think using "en" is (probably?) the only option because "sobre la costa oeste" sounds even stranger. ¦-}