Are the words "backdrop" and "background" interchangeable?
No doubt many people think of the two words as interchangeable, but to me backdrop (a US coinage from about a century ago, originally a literal reference to the backcloth of a theatre stage) potentially has much more restricted applicability.
In general, I expect a figurative backdrop to be something apparent to me (the observer) at the time when I'm viewing (or otherwise "focussing on") some foreground referent.
On the other hand, a figurative background might be something I'm completely unaware of until someone else tells me the backstory behind whatever we're talking about or looking at.
In terms of prevalence, even among people who treat the words as equivalent, background is far more common, but arguably backdrop sounds more artistic/intellectual/"cultured" to many.
from my experience, backdrop implies a sense of physicality in its definition, being used as a verb:
"lie behind or beyond; serve as a background to."
"the part of a picture, scene, or design that forms a setting for the main figures or objects, or appears furthest from the viewer."
Do you see the difference? I would say no, they are not interchangeable.
No; in some isolated contexts the words may be interchanged, but overall they have different definitions. For example, people and things can "blend into the background", but usually not into the backdrop, and there is no such thing as a "backdrop check" although most readers will be quite familiar with the concept of a "background check".