Melkor was a Vala, but after the War of Wrath the Valar denied him the title(incidentally, they did not deny the title of Maia to Sauron, Balrogs, Dragons etc).

"Morgoth"/"Dark Enemy" is just the nickname the Noldor used for him, and since the Silmarillion is written from an "Elvish" perspective, it's pretty much natural he would be named as such.

Daddy, Junior and the spook.