Can I run iOS apps on my Mac?

AFAIK, iPhone simulator doesn't download an application from the App Store; it's just used to debug an iOS application you are developing.

If you are developing iOS apps in XCode, you can use the iPhone or iPad emulator to run your app, but there is no way to run a precompiled app (such as from the app store)

Possible? Yes, but...currently the only way to run iOS apps on a Mac is via the iPhone/iPad Simulator (available in the iOS SDK)...AND you need the source code for the app you want to run.

Unfortunately the iPhone/iPad simulator is only available as part of the dev tools, and does not contain any syncing abilities with iTunes. It also doesn't have an app store app, so you can't buy applications from within the simulator.

As it's primary use is for developers to test their apps without the actual hardware you will most likely never see the features you want added to the simulator.

No it wouldn't be possible, unless somebody came a up with an iOS emulator that would let you run ARM based binaries on your Intel based Mac/PC. Basically the same as what they have done with reverse engineering the Super Nintendo and the resultant emulators that have been written to emulate Super Nintendo roms. No doubt someone will do it one day many years from now.