Given a Hilbert space \(\mathcal{H}\), the author studies noncommutative domains \(\mathbb{D}_f^\phi(\mathcal{H})\) in \(B(\mathcal{H})^n\), generated by a positive regular free holomorphic function \(f\) and some classes of \(n\)-tuples \(\phi=(\phi_1,\dots,\phi_n)\) of formal power series in (noncommutative) indeterminates \(Z_1,\dots,Z_n\). Such a domain \(\mathbb{D}_f^\phi=\mathbb{D}_f^\phi(\mathcal{H})\) has a universal model associated to the multiplication operators \((M_{Z_1},\dots,M_{Z_n})\), described via noncommutative Poisson transforms. Among several properties, all joint invariant subspaces under \(M_{Z_1},\dots,M_{Z_n}\) are given a Beurling type characterization, and the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra \(H^\infty(\mathbb{D}_f^\phi)\) is solved.