Learn Italian: Nessuno ha trovato il gattino.

Gattino is formed by taking the root of gatto (cat) and adding the diminutive suffix -ino. In Italian, adding -ino makes a noun smaller or cuter, so a little cat translates perfectly to kitten.

Would the ending of gattino change if there were multiple kittens, or if it was a female kitten?

Yes! Alteration suffixes like -ino act just like regular noun endings. For a female kitten, it becomes gattina. For multiple male or mixed kittens, it is gattini, and for multiple female kittens, it is gattine.

Why isn't there a non in this sentence to go with nessuno?

When nessuno (nobody or no one) is the subject and comes before the verb, you do not use non. You only need the double negative if nessuno comes after the verb, as in Non ha trovato nessuno (He did not find anyone).

Can I use the other diminutive suffix -etto here, like gattetto?

While -etto also means small or cute, certain words have a strong traditional preference for one suffix over another. Gattino is the standard, everyday word for a kitten, while gattetto is almost never used and would sound unnatural.