Making Cast Iron from Pig Iron
I have quoted the following answers from the UC Davis ChemWiki page on "The Extraction of Iron."


  Why is coke added as well, when the purpose is to decrease the carbon content?


Coke is used primarily as a reducing agent (see below) and comes out as carbon dioxide (and thus does not increase the carbon content of the molten iron).


  What exactly occurs in this process of making cast iron - what reactions happen in the furnace?


The reactions are given below in the narrative of the process.

The Blast Furnace

The common ores of iron are both iron oxides, and these can be reduced to iron by heating them with carbon in the form of coke. Coke is produced by heating coal in the absence of air. Coke is cheap and provides both the reducing agent for the reaction and also the heat source. The most commonly used iron ores are hematite, ($\ce{Fe_2O_3}$), and magnetite, ($\ce{Fe_3O_4}$).

The air blown into the bottom of the furnace is heated using the hot waste gases from the top. Heat energy is valuable, and it is important not to waste any. The coke (essentially impure carbon) burns in the blast of hot air to form carbon dioxide - a strongly exothermic reaction. This reaction is the main source of heat in the furnace.

$\ce{C + O_2 \rightarrow CO_2}$

The Reduction of the Ore

At the high temperature at the bottom of the furnace, carbon dioxide reacts with carbon to produce carbon monoxide:

$\ce{C + CO_2 \rightarrow 2CO}$

It is the carbon monoxide which is the main reducing agent in the furnace:

$\ce{Fe_2O_3 +3CO \rightarrow 2Fe + 3CO_2}$

In the hotter parts of the furnace, the carbon itself also acts as a reducing agent. Notice that at these temperatures, the other product of the reaction is carbon monoxide, not carbon dioxide:

$\ce{Fe_2O_3 + 3C \rightarrow 2Fe + 3CO}$

The temperature of the furnace is hot enough to melt the iron which trickles down to the bottom where it can be tapped off.

The Function of the Limestone

Iron ore is not pure iron oxide - it also contains an assortment of rocky material that would not melt at the temperature of the furnace, and would eventually clog it up. The limestone is added to convert this into slag which melts and runs to the bottom. The heat of the furnace decomposes the limestone to give calcium oxide.

$\ce{CaCO_3 \rightarrow CaO + CO_2}$

This is an endothermic reaction, absorbing heat from the furnace. It is therefore important not to add too much limestone because it would otherwise cool the furnace. Calcium oxide is a basic oxide and reacts with acidic oxides such as silicon dioxide present in the rock. Calcium oxide reacts with silicon dioxide to give calcium silicate.

$\ce{CaO + SiO_2 \rightarrow CaSiO_3}$

The calcium silicate melts and runs down through the furnace to form a layer on top of the molten iron. It can be tapped off from time to time as slag. Slag is used in road making and as "slag cement" - a final ground slag which can be used in cement, often mixed with Portland cement.

Cast Iron

The molten iron from the bottom of the furnace can be used as cast iron. Cast iron is very runny when it is molten and doesn't shrink much when it solidifies. It is therefore ideal for making castings - hence its name. However, it is very impure, containing about 4% of carbon. This carbon makes it very hard, but also very brittle. If you hit it hard, it tends to shatter rather than bend or dent. Cast iron is used for things like manhole covers, guttering and drainpipes, cylinder blocks in car engines, Aga-type cookers, and very expensive and very heavy cookware.