Ok, I have a question regarding the inquisitor and how domains work.
The inquisitor doesn't get the bonus spells.
The charm domain adds "charm person" for example. And the lvl8 power ist to cast "charm person" as a swift action.
How would this work (if it works at all) for the inquisitor?
With the domain ability, you're not casting the actual Charm Person spell. It's a spell like ability that happens to function exactly like that spell. This doesn't rely on domain spells or spell slots, regardless of what class has the domain.