Space of density matrix

Is this phrase true?

$|\psi\rangle \langle\psi| \in V \otimes V_\mathrm{dual}$

Yes, this is a correct way to see things, but the more usual view is to note that $$ V \otimes V_\mathrm{dual} \cong \mathrm{End}(V), \tag{$*$} $$ i.e., the tensor space $V \otimes V_\mathrm{dual}$ is canonically isomorphic to the vector space of endomorphisms in $V$, i.e. to the space of linear operators $\rho: V\to V$.

It's important to note that the isomorphism $(*)$ is only strictly valid in finite dimension, and that in infinite dimensionality you need to be careful with what you allow and what you don't. Thus, in infinite dimensionality, density matrices are normally required to be trace-class, positive semi-definite, self-adjoint linear operators over the system's Hilbert space. But, ultimately, the isomorphism $(*)$ is still morally true, though.

The density matrix is a representation in a particular basis of a linear operator on the Hilbert space called the density operator. This operator lives in the space of all linear operators on the Hilbert space.

NOTE: The OP asked an additional question after I answered.