The coils are difficult to remove without damaging them as they are made with some form of soft graphite (or something). They may also be glued in place prior to being soldered in places so they don't float around during the soldering process.
If you do a full replacement, what some call a windshield wipe repair, then I would apply the same temperature as usual ( I use 380C at low airflow on a Hakko 810) but make sure to heat sink the CPU and generally point your airflow away from it. You will probably want to pry the coil once the colder is flowing in that area so you don't leave heat on their longer than necessary. Practice on some dead boards.
For the rebuild, I would definitely start with the smallest components (C1513 & D1501 make sure R0412 is still there) first as you will have lots of "elbow room" to work without the big ones in place. Then it's really a matter of preference but I would keep the IC for last. The other components will help guide you for it's proper alignment and you don't heat it unnecessarily either. Keep in mind you are next to the CPU so heat transfer must be limited to what is essential.