"Chicken fried rice" is perhaps more like what might be served in Chinese cuisine, and would presumably be translated as "Nasi goreng dengan ayam"
"Nasi dengan ayam goreng" would be more like a bowl of rice with a bucket of KFC.
In fact, KFC outlets in Indonesia typically serve a pot of rice with the chicken, instead of the box of fries we get in the west. (although the fries are still an option)
'Dia' means he or she and if there are no names given then either option should be accepted. Another thing to note would be he or she can also be translated as 'dia' (formal, most common), 'ia' (informal), or 'beliau' (very formal, usually when talking about someone older than you or have authority over you like your boss for example)
I don't get how to find the with in this sentence. It would seem to directly mean, He/she wants rice meat chicken fried. How to know to add with besides memorising a posible translation seems wierd. Also, how would you know if both or only one of the items are fried? It looks to me like fried is applying to both the rice and chicken since no words actually seperate the rice from the meat.