Not just any AVR MCU you like I think. Only those supported.
Yes, of course. I didn't mean it could be any 8-bit ATMEL just that the original Arduino is based on an 8-bit ATMEL processor. Obviously you need to use a microcontroller for which support exists in the Arduino environment. The cool thing of course is that "anyone" can add support for devices to the environment which I believe is why you now can program the ESP8266 directly from the Arduino IDE.

I've not used an Arduino for anything. All I've ever done with the IDE is flashing an Openlog blackbox with its firmware.

Of course there are limitation, problems, issues and bugs - as is always the case :-)