We consider a class of recent rigidity results in a convex cone ÎŁ â R N \Sigma \subseteq \mathbb {R}^N . These include overdetermined Serrin-type problems for a mixed boundary value problem relative to ÎŁ \Sigma , Alexandrovâs soap bubble-type results relative to ÎŁ \Sigma , and Heintze-Karcherâs inequality relative to ÎŁ \Sigma . Each rigidity result is obtained here by means of a single integral identity and holds true under weak integral overdeterminations in possibly non-smooth cones. Optimal quantitative stability estimates are obtained in terms of an L 2 L^2 -pseudodistance. In particular, the optimal stability estimate for Heintze-Karcherâs inequality is new even in the classical case ÎŁ = R N \Sigma = \mathbb {R}^N . Stability bounds in terms of the Hausdorff distance are also provided. Several new results are established and exploited, including a new PoincarĂ©-type inequality for vector fields whose normal components vanish on a portion of the boundary and an explicit (possibly weighted) trace theory â relative to the cone ÎŁ \Sigma â for harmonic functions satisfying a homogeneous Neumann condition on the portion of the boundary contained in â ÎŁ \partial \Sigma . We also introduce new notions of uniform interior and exterior sphere conditions relative to the cone ÎŁ â R N \Sigma \subseteq \mathbb {R}^N , which allow to obtain (via barrier arguments) uniform lower and upper bounds for the gradient in the mixed boundary value-setting. In the particular case ÎŁ = R N \Sigma =\mathbb {R}^N , these conditions return the classical uniform interior and exterior sphere conditions (together with the associated classical gradient bounds of the Dirichlet setting).