One more thing: I may want to use a chip like the ATtiny45 for a project. Is there a way to stick with the Arduino IDE or should I learn to use AVR Studio 4, etc. for that?
The ATtiny45 will work but the ATtiny85 is a better choice. The 45 has very limited amount of SRAM and is a bit snug for Flash. But, it all depends on what you plan to do.
You will need to download a "core" and install it (basically unzip some files to the correct location). You will need an ICSP (or ISP or "programmer"). Your AVRISP mkII should work fine. You may need to use AVRDUDE directly (I haven't needed to with a Pololu programmer).