Hello,
For your first concern. All switches will need to be upgraded. However, you can run the upgrade from the master and it will push the image to all the other switches if done correctly.
To answer your second concern, let me provide a little background. On 3850 there are two modes. Bundle mode and install mode:
Bundle mode: Traditional way of installing IOS where you set the boot statement to file with the "boot system" command and reload.
Install mode: Newer method that is recommended as it optimizes memory utilization. This mode must be done using the "request platform" commands.
If you are going to run in bundle mode then your method would be correct, you would need to manually copy the image to the flash of each switch and change the boot statement and reload.
If you are going to run in install mode then the doc is correct. However, it depends on what mode you are in currently (show ver will tell you). If you are currently in install mode, follow the "Upgrading from Cisco IOS XE Denali 16.x.x to Cisco IOS XE Everest 16.6.x in Install Mode" section. If you are currently in bundle mode, follow the "Move from Cisco IOS XE Everest 16.x.x Bundle Mode to Install Mode" section. When running these commands, the files will automatically be copied to each of the member switches so you don't have to manually do it yourself.
Hope that helps!
-Bradley Selzer
CCIE# 60833