Summary: In this paper, we give a complete study on the existence and non-existence of solutions to the following mixed coupled nonlinear Schrödinger system \[ \begin{cases} - \Delta u + \lambda_1 u = \beta u v + \mu_1 u^3 + \rho v^2 u & \text{ in } \mathbb{R}^N, \\ - \Delta v + \lambda_2 v = \frac{ \beta}{ 2} u^2 + \mu_2 v^3 + \rho u^2 v & \text{ in } \mathbb{R}^N, \end{cases} \] under the normalized mass conditions \(\int_{\mathbb{R}^N} u^2\, d x = b_1^2\) and \(\int_{\mathbb{R}^N} v^2\, d x = b_2^2\). Here \(b_1, b_2 > 0\) are prescribed constants, \(N \geq 1\), \(\mu_1, \mu_2, \rho > 0\), \(\beta \in \mathbb{R}\) and the frequencies \(\lambda_1, \lambda_2\) are unknown and will appear as Lagrange multipliers. In the one dimension case, the energy functional is bounded from below on the product of \(L^2\)-spheres, normalized ground states exist and are obtained as global minimizers. When \(N = 2\), the energy functional is not always bounded on the product of \(L^2\)-spheres. We give a classification of the existence and nonexistence of global minimizers. Then under suitable conditions on \(b_1\) and \(b_2\), we prove the existence of normalized solutions. When \(N = 3\), the energy functional is always unbounded on the product of \(L^2\)-spheres. We show that under suitable conditions on \(b_1\) and \(b_2\), at least two normalized solutions exist, one is a ground state and the other is an excited state. Furthermore, by refining the upper bound of the ground state energy, we provide a precise mass collapse behavior of the ground state and a precise limit behavior of the excited state as \(\beta \to 0\). Finally, we deal with the high dimensional cases \(N \geq 4\). Several non-existence results are obtained if \(\beta < 0\). When \(N = 4\), \(\beta > 0\), the system is a mass-energy double critical problem, we obtain the existence of a normalized ground state and its synchronized mass collapse behavior. Comparing with the well studied homogeneous case \(\beta = 0\), our main results indicate that the quadratic interaction term not only enriches the set of solutions to the above Schrödinger system but also leads to a stabilization of the related evolution system.