What went wrong with my cookie dough?
Your egg and butter mix has curdled/split you need to put it in a fresh floured bowl and keep mixing till smooth.

The brown is, as you mentioned, the brown sugar. Nothing to worry about. 

Personally I wouldn't use melted butter at all. Rather I would use soft butter. Creaming sugar and butter with melted butter is impossible, especially if the butter is melted to the extent where the fat and milk have separated.

Soft bake is soft, like cake. Chewy bake is ... Chewy, like brownies.