Can I run a NAND backup before installing BootMii, and will I still have all my saves after softmodding?

As the title asks, I need to know, this is a family Wii.

No, BootMii is what performs the backup, so running it without installing it is impossible.

Also, installing BootMii and the Homebrew channel will not affect any other part of the Wii.

Well, installing Bootmii as an IOS does not affect the Wii much but installing it as Boot2 will. Boot2 is a critical part of the boot process and without it your wii will be (as usefull as) a brick