Why does cotton dissolve in a dilute solution (1/2) of fuming sulfuric and nitric acids?
To address you second question first, in neither case did the cotton dissolve.  In the diluted acid mixture the cotton was just wetted, possibly also having enough oxidation of the cotton by the nitric acid to give it a different texture.  

Regarding the effect of dilution on the nitration of the cotton, the reaction was basically just too dilute to proceed.  Also, according to this Wikipedia article, the sulfuric acid is present as a catalyst, producing the nitronium ion, $\ce{NO2+}$. The net reaction is given as:  

$$\ce{3HNO3 + C6H10O5 → C6H7(NO2)3O5 + 3H2O}$$  

Since water is a product of the reaction, by adding water to the reaction via dillution we further inhibit the formation of the products.