Context. When stripped from their hydrogen-rich envelopes, stars with initial masses between ∼7 and 11 M develop massive degenerate cores and collapse. Depending on the final structure and composition, the outcome can range from a thermonuclear explosion, to the formation of a neutron star in an electron-capture supernova (ECSN). It has been recently demonstrated that stars in this mass range may be more prone to disruption than previously thought: they may initiate explosive oxygen burning when their central densities are still below ρ c 10 9.6 g cm −3 . At the same time, their envelopes expand significantly, leading to the complete depletion of helium. This combination makes them interesting candidates for Type Ia Supernovae-which we call (C)ONe SNe Ia-and might have broader implications for the formation of neutron stars via ECSNe. Aims. To constrain the observational counterparts of (C)ONe SNe Ia and the key properties that enable them, it is crucial to constrain the evolution, composition, and pre-collapse structure of their progenitors, as well as the evolution of these quantities with cosmic time. In turn, this requires a detailed investigation of the final evolutionary stages preceding the collapse, and their sensitivity to input physics.Methods. Here, we model the evolution of 252 single, non-rotating helium-stars covering the initial mass range 0.8-3.5 M , metallicities between Z = 10 −4 and 0.02, and overshoot efficiency factors from f OV = 0.0 to 0.016 across all convective boundaries. We use these models to constrain several properties of these stars, including their central densities, compositions and envelope masses at the time of explosive oxygen ignition, and the final outcome as a function of initial helium-star mass. We further investigate the sensitivity of these properties to mass-loss rate assumptions using an additional grid of 110 models with varying wind efficiencies. Results. We find that helium-star models with masses between ∼1.8 and 2.7 M are able to evolve onto 1.35-1.37 M (C)ONe cores that initiate explosive burning at central densities between log 10 (ρ c /g cm −3 ) ∼ 9.3 and 9.6. We constrain the amount of residual carbon retained after core carbon burning as a function of initial conditions, and conclude that it plays a critical role in determining the final outcome: Chandrasekhar-mass cores with residual carbon mass fractions of X min ( 12 C) 0.004 result in (C)ONe SNe Ia, while those with lower carbon mass fractions become ECSNe. We find that (C)ONe SNe Ia are more likely to occur at high metallicities, whereas at low metallicities ECSNe dominate. We constrain the relative ratio between (C)ONe SNe Ia and SNe Ib/c to be 0.17-0.30 at Z = 0.02, and 0.03-0.13 at Z ≤ 10 −3 . Conclusions. We conclude with a discussion on potential observational properties of (C)ONe SNe Ia and their progenitors. In the few thousand years leading to the explosion, at least some progenitors should be identifiable as luminous metal-rich supergiants, embedded in hydrogen-free circumstellar nebulae.