1) Mario Kart works just fine.
2) I believe most Wii Modchips, or at least some don't modify the firmware of the wii (hence why you can update and still play) but actually mod the DVD drive.
3) I believe it would be legally dubious, I would even guess that releasing software within a game package to render mod chips, but not Wii's inoperable, i.e. no longer function, would be legally dubious as well.
Having said that:
Nintendo - News - Information regarding mod chips
| Nintendo - Corporate Information | Legal Information (Copyrights, Emulators, ROMs, etc.)
Nintendo Supports U.S. Raids on Copyright Circumvention Devices at Nintendo :: What's New
So it looks like it depends on where you live. As end users of mod chips i'm not sure where we legally stand.
I'm not sure how they would detect the chip and what legal rights Nintendo have to block your Wii - After Mario Galaxy bricked Wii's there was talk of people sending them in and having units repaired by nintendo without any consequence to their mod chip.
The IOS37 is yet to be activated - all though for the first couple of days people were convinced it was in Mario Kart. IOS37 will not as i understand prevent you from playing backups but will prevent software such as freeloader from operating as i understand it. Therefore restricting you to using your own region games.
Backups are dubious as well - they seem to have been removed from being mentioned in most software licenses, or are expressly forbidden. The second copy rule does not always apply - it used to but changes in the law put an end to it, again this may vary from region to region. From what i understand if you create your own back up that should be OK if you download or have passed to you a back up that is where things become dubious. I know Microsoft actually say you can no longer make back ups and without a Wii manual to hand I can not be sure.
So back on point:
Mario Kart will not screw up a modded system, and i think it's a bluff on Nintendo's part to put people off, i may be wrong, as i doubt Nintendo really want to spend the money to tighten the security in their system as it costs an increasing amount each time.
However if Nintendo found a non permanent solution, in the coding of the games operating for that game only and not your other backups, to prevent a backup version from being usable then that would be perfectly legal as far as i can see.