1)
Was it called Eretz Edom even before Esav conquered it?
1.
Ha'amek Davar #1: It is called Eretz Edom now, for in the future it will be Eretz Edom.
2.
Ha'amek Davar #2: It was already called Eretz Edom, because its soil was red (Bereishis Rabah 75:4).