If you get a base with CX (anywhere other than HKG) they do not help you with immigration, you have to be legal to work in the country you chose to be based in. You cannot come and go as you please working as a based pilot as are rostered out of the country you are based in. USA you need to be totally legal to work there. You could however commute to HKG to work but live wherever you want pretty much. A lot of people do that but not many from the states as quite a long way, but it is done. No idea about other airlines. Also if you are not hired as a DEFO onto a base you will wait at least 5 years before you can apply for a base probably longer.