For finite interacting particle systems with strong repulsing-attracting or general interactions, we prove global weak well-posedness almost up to the critical threshold of the strengths of attracting interactions (independent of the number of particles), and establish other regularity results, such as a heat kernel bound in the regions where strongly attracting particles are close to each other. Our main analytic instruments are a variant of De Giorgi's method in $L^p$ with appropriately chosen large $p$, and an abstract desingularization theorem.