I think you need to clarify what you’re talking about when you say “inflation” and “physically detected.” If you’re talking about the expansion of the universe, then yes, that’s been detected. We can see from Dopper shifts that some galaxies are moving away from us, and we can see that galaxies that are farther away are moving away faster. If by “inflation” you mean the period of rapid cosmic expansion shortly after the Big Bang, I’m not sure that that’s detectable in but it’s theorized. I think I would say that dark matter and dark energy have both been detected. We can see their effects. We don’t know what exactly they are, which is why they have a super cool name that makes regular matter look lame by comparison.
No, they’re not. The Big Bang Theory existed before either of these concepts, and the first piece of support for that theory is the cosmic microwave background. The Doppler shifting of light is also a significant piece of evidence.