"The grey cat has an orange fish."

Yes, someone else suggested in "Grå katten" the color would be the name of the cat as in "Gray the cat".

de is used for plural things, so it would be "de grå katte" (the grey cats), but "den" is used for singular common gender words like here. "det" is used for neuter gender singular words ("det grå hus" = "The grey house")

Because the word is "orange". Adjectives ending in "-e" also tend not to take a separate neuter or plural forms:
"En orange fisk" = An orange fish
"Et orange hus" = An orange house
"Orange biler" = Orange cars
"Den orange fisk" = The orange fish

Because one is a colour and the other is a fruit. :)
The word orange is French, coming via the Spanish naranja from Arabian nāranǧ, which refers to the bitter orange fruit. The French referred to it as pomme d'orange, "orange apple", too, so there's the connection to the colour.
Appelsin, on the other hand, is a common name in central Europe, originated in older Dutch as appelsina. The fruit was imported from South China and the name simply means "apple from China".

The gray the cat? :)

No, if you add an adjective, you split the article from the noun again because the adjective needs to squeeze in between the two:
kat -> katten -> den grå kat