Bootmii is an IOS (kinda), it gets installed to IOS 254. It can also be installed to boot2 in older Wii's. The private folder is what holds all the official Wii stuff, like copied save games and message board data. If you already have the HBC, you can just run the modmii wizard and when it asks if your Wii is a virgin say no. Even though you only have bootmii as IOS, I would still highly recommended making a backup, just to be safe. The new wiimote+ remotes not working is a known issue. Some homebrew has been updated with a fix, other homebrew has not.
The private folder files are just data copied from the main Wii memory. Message board data: I've never heard of that being stored on the SD card. Maybe under the Photo Channel data, which can be transferred to the SD card. I heard the new motion+ remotes weren't working, but I didn't have a problem with them. They worked with both my 2008 and '10 Wiis, even in the Homebrew Channel menu and all its subsidiaries. I couldn't understand why people were having problems with them; maybe they didn't have all the updates installed. I think I might try installing Bootmii on 254 since it's a black Wii...