In a Non-Orthogonal Unicast and Multicast (NOUM) transmission system, a multicast stream intended to all the receivers is superimposed in the power domain on the unicast streams. One layer of Successive Interference Cancellation (SIC) is required at each receiver to remove the multicast stream before decoding its intended unicast stream. In this paper, we first show that a linearly-precoded 1-layer Rate-Splitting (RS) strategy at the transmitter can efficiently exploit this existing SIC receiver architecture. By splitting the unicast messages into common and private parts and encoding the common parts along with the multicast message into a super-common stream decoded by all users, the SIC is better reused for the dual purpose of separating the unicast and multicast streams as well as better managing the multi-user interference among the unicast streams. We further propose multi-layer transmission strategies based on the generalized RS and power-domain Non-Orthogonal Multiple Access (NOMA). Two different objectives are studied for the design of the precoders, namely, maximizing the Weighted Sum Rate (WSR) of the unicast messages and maximizing the system Energy Efficiency (EE), both subject to Quality of Service (QoS) rate requirements of all messages and a sum power constraint. A Weighted Minimum Mean Square Error (WMMSE)-based algorithm and a Successive Convex Approximation (SCA)-based algorithm are proposed to solve the WSR and EE problems, respectively. Numerical results show that the proposed RS-assisted NOUM transmission strategies are more spectrally and energy efficient than the conventional Multi-User Linear-Precoding (MU-LP), Orthogonal Multiple Access (OMA) and power-domain NOMA in a wide range of user deployments (with a diversity of channel directions, channel strengths and qualities of channel state information at the transmitter) and network loads (underloaded and overloaded regimes). It is superior for the downlink multi-antenna NOUM transmission.