The correct statement is:

We have to decide which one of the four given statements is factually true in the light of standard metallurgical principles. One by one we shall examine every option and compare it with the well-known facts of metallurgy, showing all the chemical equations wherever necessary.

First, look at option A:
“Pig iron is obtained from cast iron.”

The blast furnace reduction of haematite $$\text{Fe}_2\text{O}_3$$ with coke gives a crude iron melt containing about $$4\text{-}5\%$$ carbon. This first product that runs out of the furnace and solidifies in moulds is called pig iron. Cast iron, on the other hand, is produced later by melting pig iron again along with scrap iron and adding controlled amounts of carbon and silicon. Thus the chronological order is
$$\text{haematite}\;\longrightarrow\;\text{pig iron}\;\longrightarrow\;\text{cast iron}.$$

So pig iron is obtained before cast iron, not from it. Therefore option A is incorrect.

Now examine option B:
“The Hall-Héroult process is used for the production of aluminium and iron.”

The Hall-Héroult electrolytic cell contains a fused mixture of cryolite ($$\text{Na}_3\text{AlF}_6$$) and alumina ($$\text{Al}_2\text{O}_3$$). At about $$950^{\circ}\text{C}$$ the cell supplies aluminium metal at the cathode by the reaction

$$\text{Al}^{3+}+3e^{-}\;\longrightarrow\;\text{Al(l)}.$$

No iron species are present in the bath, and the process is never used for iron. The major industrial method for iron is the blast furnace. Hence option B is also incorrect.

Next consider option C:
“Leaching of bauxite using concentrated NaOH solution gives sodium aluminate and sodium silicate.”

During the Bayer (leaching) process, powdered bauxite is digested with hot concentrated $$\text{NaOH}$$ under pressure. The principal reactions are

$$\text{Al}_2\text{O}_3 + 2\,\text{NaOH} + 3\,\text{H}_2\text{O} \;\longrightarrow\; 2\,\text{Na[Al(OH)}_4]$$

$$\text{SiO}_2 + 2\,\text{NaOH} \;\longrightarrow\; \text{Na}_2\text{SiO}_3 + \text{H}_2\text{O}.$$

The first product $$\text{Na[Al(OH)}_4]$$ is called sodium aluminate, and the second product $$\text{Na}_2\text{SiO}_3$$ is sodium silicate, exactly as the statement claims. Therefore option C is correct.

Finally analyse option D:
“The blistered appearance of copper during metallurgical process is due to the evolution of CO$$_2$$.”

In the final roasting of the matte, impure molten copper absorbs dissolved $$\text{O}_2$$. When the metal solidifies, excess oxygen combines with dissolved sulphur to form $$\text{SO}_2$$ gas:

$$\text{Cu}_2\text{S} + \text{O}_2 \;\longrightarrow\; 2\,\text{Cu} + \text{SO}_2\uparrow.$$

The escaping $$\text{SO}_2$$ creates cavities that give the solid ingot a blistered or vesicular surface. Carbon dioxide $$\text{CO}_2$$ is not involved. Hence option D is incorrect.

Out of all four statements, only option C withstands critical scrutiny.

Hence, the correct answer is Option C.