Will this simulate the sun and planets of the solar system?

Do these models of n-body gravity predict the perihelion in the orbit of Mercury?

Newton's does not predict perihelion, GR General Relativity does, Fedi's SQG Superfluid Quantum Gravity with Gross-Pitaevskii does, and this model of gravity fully-derived from the Standard Model also predicts perihelion in the orbit of planet Mercury.

Lagrange points like L1 and L2 are calculated without consideration for the mass of the moon.

Additional notes on n-body mechanics: https://westurner.github.io/hnlog/#comment-45928486 Ctrl-f n-body, perihelion