quad boot MacBook Pro (Retina, 15-inch, Mid 2014) -- will 10.11.6 -> 10.12 affect boot/partitions/etc?
You do not have Window Recovery Environment partition so you probably deserve to get clobbered during your next major Windows 10 update. Also, there is no Microsoft Reserved partition, but that is not unusual on Macs.

Installing Linux Mint without a swap partition seems like a dangerous move.

I would assume gdisk would display type 8300 and 8200 for linux and swap. You show the legacy value of 0700.

I assume you using GRUB as your boot manager. I would recommend the Mac built in Startup Manager. This would require multiple EFI partitions. Or a better alternative would be multiple small HFS+ partitions, each with a silent rEFInd installed.

Before any major upgrade, I suppose backing up the values in the GPT is a good idea. You could use the gpt command to write the GPT to the EFI partition as a text file. Also, I suppose you could copy the files in the EFI to a dmg file.

An upgrade fro 10.11.6 to 10.12.x would probably not present any major problems. Upgrading to 10.13.x can cause issues if your /dev/disk0 is a SSD. It has been reported that the macOS boot partition is converted to APFS when /dev/disk0 is a SSD.

Also, keeping a USB Windows 10 installer handy is a good idea. You never know when a change to hardware or software to require a rebuild of the BCD file.

If you need further clarification, please comment.