The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure or dimension, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy (MID) property holds, i.e., a sufficiently high multiplicity implies that the root is dominant, leading to designs based on directly assigning multiple roots. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback with two terms in the control. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots, and the property of corresponding to (local/global) minimizers of the spectral abscissa function.