We do have our share of misunderstandings and misapprehensions here.

From my reading ...
Allotropic (pure) iron is soft and weak, compared to most alloys. It is the hardest and strongest of the pure metals.
Wrought iron is what you get when you smelt iron ore. Also known as pig iron. It's the raw material for cast iron and steel.
Steel is either wrought iron heated in charcoal, or the modern alloy made by adding carbon to molten iron. The modern stuff is much better. Though it should be noted that true steel needs manganese as a catalyst for proper alloying. However, too much manganese turns the steel brittle. Something the Italians ran into during World War II (Italian armor plate would crack when hit by an anti-tank shell).
Cast iron is made by adding carbon and silicon to molten iron, and then casting it. Cooled fast it becomes black iron, cooled slowly white iron. Gray iron is cast iron that cooled at a normal rate.
Note that metal can be smelted without melting it. All you need to do is heat it until it becomes plastic. At which time material with a different melting point will separate out. But not entirely, which is how you get slag. Cast metals are stronger than wrought metals by and large, while die cast metals are stronger yet. However, die casting involves very high pressure and very small lots. For examples of die cast metal check the change in your pocket.
The whole 'cold iron' thing? It's what you get when a clueless academic tries shrouding his tush. It's of a type as the idea Chuchulain's gaer bolg (foot spear) was some sort of spiky soccer ball. It's iron that burns the fey; gives them an incentive to learn well the arts of war. (He can't hit you, he can't hurt you.)