I noticed that some of Japan Wii games are missing languages. I'm sure the vast majority of these are just "Japanese", but the ones without language flags should probably be checked.

What's the process for language checking these using Dolphin?

I thought the Japanese Wii didn't have any other languages in the BIOS settings, so I assume any language checking would be through in-game menus. Can someone confirm?

Yes, in-game menus is the correct way to check them. Note that some games will crash Dolphin if you have it set to English. Here's one that crashed for me: http://redump.org/disc/29037/