I always edge and trim first. If the drive, sidewalk, or patio have debri on them I will go ahead and blow the debri into the yard so that it get mulched in and does not show when done. I may have to blow everything off one more time when done, but it is minimal and real quick. This way the final results look better.
IMO, edging house and sidewalks first is best. If you mow first, then go back to edge the tall grass left it will just lay on lawn. By edging then mowing, you are mulching up that grass and it mixes with lawn. Leaving a clean finished cut. Then I finish with blowing grass off drive and sidewalks. Then if any grass in street I blow it off street. Imo it looks bad to spread the grass across street. Unprofessional.
Huh? Going to the blower several times? You can take a mower down the sidewalks or driveway. You can use the trimmer as a kind of blower on the patios to blow the large stuff into the turf area to be mulched.
Anyway, most of the time its edge and mow first. The mower is a lot faster and more productive than a trimmer. With the mower I can get within about a 1/2" of everything. Then the trim time is very little because instead of have to cut a lot of grass with the trimmers its just trimming a little. You also have a guide on height.