Why does cotton dissolve in a dilute solution (1/2) of fuming sulfuric and nitric acids? A nitration experiment carried out showed that a solution of half diluted sulfuric and nitric acids would turn the submerged cottons into a gooey substance. The remnants of the cotton balls did not flash as nitrocellulose would have.

On the other hand, cotton balls submerged in a mixture of fuming nitric acid and fuming sulfuric acid remained intact and were nitrated, producing nitrocellulose.

What I'm really curious and confused about is how the dilution not only interferes with the nitration itself, but also affects its dissolution. 

Any guidance is appreciated.
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.