Learning safe and stable robot motions from demonstrations remains a challenge, especially in complex, nonlinear tasks involving dynamic, obstacle-rich environments. In this letter, we propose Safe and Stable Neural Network Dynamical Systems S<inline-formula><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula>-NNDS, a learning-from-demonstration framework that simultaneously learns expressive neural dynamical systems alongside neural Lyapunov stability and barrier safety certificates. Unlike traditional approaches with restrictive polynomial parameterizations, S<inline-formula><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula>-NNDS leverages neural networks to capture complex robot motions, providing probabilistic guarantees through split conformal prediction in learned certificates. Experimental results in various 2D and 3D datasets—including LASA handwriting and demonstrations recorded kinesthetically from the Franka Emika Panda robot—validate the effectiveness of S<inline-formula><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula>-NNDS in learning robust, safe, and stable motions from potentially unsafe demonstrations.