The effect of concentration on the rate of a homogenous reaction
I've also read online that adding $\ce{H2O2}$ and $\ce{H2SO4}$ creates a &quot;Piranha&quot; solution [...]

This is true - in principle. In reality, Piranha solution is made from concentrated sulfuric acid and 30% hydrogen peroxide.
In the case of your experminent, you will most likely use diluted sulfuric acid and aqueous solutions of hydrogen peroxide with a much lower concentration. Consequently, the Piranha effect will not play a role in your experiment, neither is the oxidizing power of sulfuric acid relevant here.
On a side side note, treating a sample with conc. $\ce{H2SO4}$ is a method for the qualitative determination of iodide: Upon heating, one can observe the violet vapour of iodine.
In your case, you need the sulfuric acid to keep the pH value low. Under these conditions, $\ce{H2O2}$ acts as an oxidant:
$$\ce{H2O2 + 2 H+ + 2 e- -&gt; 2 H2O}$$

Does dissolving into ions also count as a reaction?

No.

Does the Starch (for which I have no given chemical formula) react in solution A?

Starch is a mixture of two natural polymers (amylopectin and amylose), both made from D-glucose. Amylopectin doesn't play a role here, but it doesn't disturb either and it's just cheaper to let you work with the natural mixture.
It's the helical amylose that is relevant here - and it acts as a reagent.
What actually happens in the reaction is the following:
$$\ce{2 I- -&gt; I2 + 2 e-}$$
$$\ce{I2 + I- -&gt; I3-}$$
The linear triodide ion forms a strongly coloured inclusion complex with the amylose in the starch.