Identify the reactants and product of the reaction.

The introduction states that iron (steel wool) reacts with oxygen to form iron oxide ($Fe_2O_3$).
Reactants: Iron ($Fe$) and Oxygen ($O_2$)
Product: Iron(III) oxide ($Fe_2O_3$)

To balance the oxygen atoms, multiply $O_2$ by 3 and $Fe_2O_3$ by 2:

Now, balance the iron atoms. There are $2 \times 2 = 4$ iron atoms on the right side, so multiply $Fe$ by 4:

The balanced chemical equation for the reaction is $\boxed{4Fe(s) + 3O_2(g) \to 2Fe_2O_3(s)}$.

Iron oxide ($Fe_2O_3$) is a metal oxide. Metal oxides are generally basic in nature. When dissolved in water (as described in step 9 of the procedure), it will form a basic solution. Basic solutions turn red litmus paper blue.
The expected color change for the litmus paper is $\boxed{\text{red litmus paper turns blue}}$.