What about "Jeg er bra"? Could it be used as a translation of "I am doing well"?
How about "Jeg har det bra." ?
This is what most people in Norway say
I have it good!
Can you use 'gjøre' in this context?
No. "Jeg gjør det godt" = "I am performing well"
"Jeg gjør det godt" would most often mean you are getting good return on investments or you have a profitable business. Another possibility would be that you are doing well in your studies.
So if you wish good upon someone, you can say "Ha det godt!"?
Ha det godt or Ha det bra, both are fine
"Det har jeg godt" would make any sense?
No. "Jeg har det bra."