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We present the phase diagram, the underlying stability and magnetic properties as well as the dynamics of nonlinear solitary wave excitations arising in the distinct phases of a harmonically confined spinor $F=1$ Bose-Einstein condensate. Particularly, it is found that nonlinear excitations in the form of dark-dark-bright solitons exist in the antiferromagnetic and in the easy-axis phase of a spinor gas, being generally unstable in the former while possessing stability intervals in the latter phase. Dark-bright-bright solitons can be realized in the polar and the easy-plane phases as unstable and stable configurations respectively; the latter phase can also feature stable dark-dark-dark solitons. Importantly, the persistence of these types of states upon transitioning, by means of tuning the quadratic Zeeman coefficient from one phase to the other is unravelled. Additionally, the spin-mixing dynamics of stable and unstable matter waves is analyzed, revealing among others the coherent evolution of magnetic dark-bright, nematic dark-bright-bright and dark-dark-dark solitons. Moreover, for the unstable cases unmagnetized or magnetic droplet-like configurations and spin-waves consisting of regular and magnetic solitons are seen to dynamically emerge remaining thereafter robust while propagating for extremely large evolution times. Interestingly, exposing spinorial solitons to finite temperatures, their anti-damping in trap oscillation is showcased. It is found that the latter is suppressed for stronger bright soliton component "fillings". Our investigations pave the wave for a systematic production and analysis involving spin transfer processes of such waveforms which have been recently realized in ultracold experiments.