PPRuNe Forums - FAA/EASA questions

Hi everybody,
My name is Joseph and I'm a aeonautic engineer from Barcelona, Spain. During these last days I'm doing a document about certification of new avionics equipment so I've been searching for some information about EASA and FAA method of certification, I've visited FAA,JAA,EASA,RTCA,EUROCAE websites and I think I have a little general scheme of the process:

The aircraft must obtain the Type Certificate
Then the Airworthiness certificate
Operational approval

In the case of any equipment update:

Equipment must follow the corresponding TSO/ETSO
Apply for TC/change of TC/STC
Installation approval
Airworthiness certificate
Operational approval

I've some doubts about this scheme so, if someone could give me some help or information I would be very grateful. I also have another question: if an aircraft has been approved with EASA regulations (so its equipment) and decides to fly to USA, there are any limitations of regulations?and if it was the contrary case?
Thank you very much in advance.