Honestly, the only right answer is that you plug it into whichever amp lets you get closest to the sound that you want to hear and to fuck with all the rules!
That said, I tend to think of the Pod as just what you said - it's an amp, a speaker, and a mic. Now, if you were to just take an amp on stage, plug it into a speaker, and put a mic in front of the speaker, what would you plug that mic into? Would you plug it into another guitar amp, or the PA? And where would you plug in the pedals?
The synth is kind of a different question. They almost always either run to a full range amp (a good keyboard amp is essentially a PA) or direct to the FOH PA, because again the sound is assumed to have been shaped to exactly what we want already, and we just want to make it louder. There's no reason you can't run it through a guitar amp (real or modelled) if you want to shape the sound further, or in ways that you can't recreate on the synth itself.
Now, having said all that, some amp modellers put out a kind of "digital hash" or noise or distortion that sits and spits and fizzes up in the frequency range that a real guitar speaker won't pass. They seem to be getting better all the time, but it seems to be the biggest real complaint I've heard about the things. A simple low pass filter somewhere around 8K usually helps this immensely. Even just turning down the "Treble" knob on the PA channel helps quite a bit. Those high frequencies aren't really what give the cabinet model its signature, anyway. We're interested mostly in the midrange resonances that all guitar cabs have. And that's why we usually want to run a modeller into a flat amp/speaker cabinet. Run it through a guitar amp and you'll have its midrange resonance superimposing over whatever the modeler is trying to do. That might sound cool, but it won't sound like the cabinet that you called up on the unit.