Back in "the day" oil was standard practice for trim but these days it is unusual for oil to be used, especially on new work. There are several reasons, number one is the fact that oils are becoming extinct due to the EPAs VOC regs. There is also coating compatibility issues if you spray the trim and cut back with the latex wall paint.... and you will need two rigs for spraying, one for latex and one for oil.
As far as method for spraying , there are many but the doors are usually removed from the hinge pins, numbered and garage sprayed. The walls are masked and the trim is sprayed, if you use an acrylic trim paint and are pretty good with a spray gun masking may not be required if you are spraying the trim before finishing the walls.
It is impossible to touch up sprayed walls that have not been back rolled, most pros apply the final coat on the walls with a standard "cut and roll" method.
I would suggest that you check out some high end acrylic enamels for the trim...