If you do a Wii System Update can anything bad happen, like something getting deleted?

It shouldn't. You can always contact Nintendo to be sure though...

Only if you have something on your wii that shouldn't be there in the first place like the homebrew channel, but I don't think that gets deleted.

Before, if you had Homebrew software on the Wii after a firmware the machine could brick, to avoid this problem Nintendo made their update system so that it deletes homebrew and foreign software from the Wii Memory to prevent the system from bricking. If you're scared about losing game saves, no that won't happen.

I've performed a couple, and I've never lost anything

As long as you're not dicking with the firmware by loading a different region menu or something goofy like that you shouldn't have a problem.

This isn't the PS3 ya know! ;-P

Thanks, I just wanted to know, just in case.

I got another question.
There is a phone service called Embarq, and I have one of their basic broadband Internet modems, does anyone know if one of those would work with a Wii LAN Adapter?

As long as the modem uses an ethernet cable rather than a USB cable, then it should be fine.

O.K., thanks.