Your "in all systems of interest" is very nearly the same as "in all systems that occur in nature", which you might find an acceptable answer to your first question. Hamiltonian/Lagrangian systems are a subset of all Newtonian systems. Newtonian systems that have nonconservative force laws are of interest, but Poincaré is arguing in a fundamentalist vein with all degrees of freedom explicitly included.
In the early 20th Century, chemists were using the atomic hypothesis freely, but there was still no robust proof of the existence of atoms as opposed to there being a system of useful empirical rules. Precise measurements and the atomic explanation of Brownian motion was one of the tipping points, but at least until that time philosophers of an empirical bent denied the existence of atoms. There were some significant holdouts until the 1910's. Whether the chemist's atoms were point-like or indivisible was not especially of practical importance.
Energetics is more than just Hamiltonian/Lagrangian physics; it is also a part of a thermodynamic approach to nature, without taking statistical mechanics as an explanation for the thermodynamics. There can be bulk thermodynamic properties such as temperature and the associated heats without requiring an explanation, whereas Newtonian mechanics, not having a fundamental concept of heat, can only(?) explain heat as a statistical property of large numbers of very small objects that (mostly, because chemistry) don't break apart.
I suggest you try the Stanford Encyclopedia of Philosophy article http://plato.stanford.edu/entries/atomism-modern/, perhaps particularly Section 5.4 (SEP is generally a pretty good online resource).