Collisions in classical mechanics

The only force really considered in classical mechanics is gravity. And yet, we often have problems involving collisions and friction, which are intrinsically electrical phenomena, and thus outside the scope of classical mechanics. We have laws such as conservation of momentum which is used for collision problems, but in the end this is all based on the assumption that collisions obey Newton's laws (the 3rd in particular). Also, including collisions and friction basically means accepting the electrical force, and thus Maxwell's equations, which have been shown to contradict classical mechanics (galilean invariance vs lorentz invariance).

Including friction and collisons also causes the systems to be hard to describe using Lagrangian or Hamiltonian approaches (discontinuous potentials), which is another argument that they might be "out of place" in mechanics.

And yet, any course on mechanics has a section on collision problems and friction force. Can we really consider them as classical mechanics phenomena? How would the forces causing collisons be described in a consistent way with Newton's laws?

Would mechanics be consistent if we assumed collisions did not exist (i.e. objects pass through each other) and there was no friction? Even rigid bodies would not exist then...