This is definitely too simple of a characterization. It's even false for n=3, before the really nasty stuff caused by non-smooth Schubert varieties shows up.

Counter-example: Consider the Soergel bimodule $R\otimes_{1,2} R\otimes_{2,3} R$, and twist the right action by the transposition $(1,2)$. Thought of as a coherent sheaf on the product $\mathbb C^n\times\mathbb C^n$, this is supported on the graph of the symmetric group elements $1, (12), (312), (321)$. The fact that it has support on the diagonal shows that there are invariant vectors, and your first two conditions are unchanged when you flip the action, as you noted.

On the other hand, the group elements that a Soergel bimodule is supported on the graphs of are an ideal in Bruhat order (an instant consequence of the same fact for fixed points in Schubert varieties), which the elements I listed above are not.

As a general comment, I think Soergel bimodules are really special, and you will need something much more powerful than conditions like the ones you've listed to describe them. I would actually be pretty surprised to see a clean characterization along these lines.

EDIT: Just as an extra comment; if $\overline{BwB}\subset GL_n$ isn't smooth, then $IH_{B\times B}^*(\overline{BwB})$ (equivariant intersection cohomology) is a Soergel bimodule and $H^*_{B\times B}(\overline{BwB})$ (usual equivariant cohomology) is not. I will believe that a combinatorial characterization is plausible when I can see why it allows the former and rules out the latter, but frankly I have no idea how that's going to happen.

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This is definitely too simple of a characterization. It's even false for n=3, before the really nasty stuff caused by non-smooth Schubert varieties shows up.

Counter-example: Consider the Soergel bimodule $R\otimes_{1,2} R\otimes_{2,3} R$, and twist the right action by the transposition $(1,2)$. Thought of as a coherent sheaf on the product $\mathbb C^n\times\mathbb C^n$, this is supported on the graph of the symmetric group elements $1, (12), (312), (321)$. The fact that it has support on the diagonal shows that there are invariant vectors, and your first two conditions are unchanged when you flip the action, as you noted.

On the other hand, the group elements that a Soergel bimodule is supported on the graphs of are an ideal in Bruhat order (an instant consequence of the same fact for fixed points in Schubert varieties), which the elements I listed above are not.

As a general comment, I think Soergel bimodules are really special, and you will need something much more powerful than conditions like the ones you've listed to describe them. I would actually be pretty surprised to see a clean characterization along these lines.