Yes, it can run on an unmodded wii.
Yes, you do have to re-run it unless you get the homebrew channel (the one with no limits). If you do get the homebrew channel, you just need to start it like any other wii channel and it will come up with a menu of all of your homebrew.
Currently, it is unknown what will happen over time with firmware updates, but as of right now there is no real risk. I do recommend checking back here or on other forums before updating your wii though, because nintendo might add something that will brick your wii.
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