For ODEs, the standard theorem of continuous dependence of initial parameters deals only with functions that are Lipschitz. Do there exist more general results holding for non-Lipschitz functions? If so, could someone direct me to some resources on the topic?
The continuous dependence on initial conditions and parameters (and even on the right-hand side in the compact-open topology) is a consequence of the uniqueness. See Theorem 3.2 of Chapter II in Hartman's "Ordinary differential equations" (I call this statement the Kamke lemma). Note that without the uniqueness, the question of continuous dependence (in the classical sense) is incorrect. So when you have this correctness, you automatically have the continuous dependence.