Bear, The ol' FAA 337 - wherein one must
Complete the form as instructed in this AC, excluding item 3, and ensure that items 6 and 7 have been properly executed. Give a copy of the form to the aircraft owner/operator and send a duplicate copy to the FAA Aircraft Registration Branch within 48 hours after the airframe, aircraft engine, propeller, or appliance is installed on an aircraft and approved for return to service.
Now which A380 sports an American N registration? Bueller, Bueller ....Anyone?
The answer to your last question is commonly known as
MOD C (all 4 engines)- no restrictions to use of.