Can I play music on my 2005 Toyota Corolla through my IPhone?

Or do I have to buy a different stereo?

You need a new stereo unless you have a casette player. If you have a casette player you can go to walmart and buy one of those mp3 to casette converters. Its a casette player with a wire coming out that plugs into your iphone. Otherwise, yes, you need a new stereo.

I find it unlikely that a Toyota from 05 would have a bluetooth compatible anything.

If it has bluetooth on it which I don't think it does, you gotta get it installed. Or else you just have to plug the thingamajigy in the hole and BAM!

Yes. Buy a aftermarket radio and it'll work perfectly fine! Go for one that has usb, bluetooth, and aux or 1/8 in jack