<p>I'm all for language diversity, but Objective C is insane. So I'm curious: is it possible to code iPhone apps with C++ while using the Cocoa API, etc?</p>

Short answer, yes, sort of. You can use Objective-C++, which you can read about at
Apple Developer Connection.
If you know C++ already, learning Objective-C would be pretty simple, if you decided to give that a try. More info on that topic is at the ADC as well.