Some questions answered

"If 2 systems are allowed to exchange E (but not N or V) heat 'flows'. But how does it actually flow? In other words, how is energy exchanged microscopically? Via collisions? Via rearrangement of potential energies?"

This is exactly how energy moves around from one place to another. Molecules are constantly bombarding each other. Each molecule exerts a potential energy "field" around it which is felt by its neighbors. Even in dilute gases, molecules occasionally whiz past each other long enough to feel these potential energies, or occasionally even get close enough to be involved in what might be considered a collision (it's hard to define rigorously exactly when a collision occurs, since in reality, most atoms are kind of 'soft'). It is through these interactions that molecules exchange kinetic and potential energies.

Imagine, in the simplest scenario, that someone breaks a racked set of billiard balls. In that case, the kinetic energy of the cue ball is divided up and transferred to the kinetic energies of all the other balls. This is not all that different than bringing a molecule at a high temperature in contact with molecules at a lower temperature.

"What is the fundamental definition of a phase? In other words, what is different (or the same) between a system of 2 phases and a situation where two systems are just sitting next to each other?"

You're anticipating our discussion of phase equilibrium later on! Basically, a phase is a homogeneous system: no spatial variations in density, energy, temperature, etc. We can consider two phases in equilibrium as being a subset of the situation where two systems sit next to each other. The idea of phase equilibrium, however, is that these two phases sit next to each other on their own, without barriers, membranes, etc. That is, water spontaneously separates into a phase of ice and a phase of liquid at 0 degrees C.