> Does an atom displace spacetime?

No.

> Is spacetime between the nucleus and the electrons?

There is space between them, i.e. they have some distance to each other (ignoring some technical details from quantum mechanics). That applies to all times, so you could say that there is "spacetime between them", but I don't think that's a useful way to view it. The same applies to all extended objects, including nuclei.

> Or is spacetime right beside me when I'm sitting in my living room?

Is "beside you" a place? Yes. That's part of space, which is a part of spacetime.