Can I start the engine without the advanced key?

If it has a transponder then you can. Valet key can be used to start but it would not open the boot.

Well, I have inserted the key (it is not the auxiliary key found in the advanced key fab, but rather a normal sized metal key with large black plastic top) into the ignition switch and the switch will not turn. I don't understand. :(

The steering could be locked. Try rocking the steering wheel gently, turning it away from the side that does move and turn the key.

Yup, tried that too. :-/

Is it possible the key isn't programmed or something? I've never used it. It does lock and unlock the doors...

Even if it is not programmed, you should be able to turn it.

Well, I guess I'm screwed. Ugh.

Try inserting the key again and ensure it is fully slotted in. While turning the key, try wiggling it slightly and also rocking the steering wheel as well.

Try putting your foot on the brake pedal.