I would really argue that "electromagnetic stuff" is what "touches" means in the first place.

Atoms aren't really empty. They're electron clouds with an extremely dense core of protons and neutrons, but the electron clouds are what we care about.

I think it's really hard to define this fully. The most relevant parts are probably the Pauli exclusion principle (which prevents two fermions from occupying the same point in space if they are otherwise identical), and the size of elementary particles (which, as far as we can tell, is 0). Overall this means that you can get two fermions arbitrarily close, but not to overlapping positions. Since they have no know dimension, they can't "touch" unless they are completely overlapping, and since they can't overlap, and since the electromagnetic repulsion increases rapidly as you get closer and closer together, I think the parent poster is right - regardless of inertia, they will always have some kind of electromagnetic interaction that will change their course or nature before actually "touching".

To add a bit of a complication, if two particles that have mass, even one as tiny as the electron, come closer together than some minimum, general relativity predicts that they would collapse into a (really really tiny) black hole. Of course, we don't know how and if general relativity really works at this level, so the relevance of this is unclear.

No, it's just different field interactions. None of this is tangible in the way that billiard balls are tangible. With particle physics there are no "things" that can "touch" - just different events in different particle fields which are more or less likely at different energies and spatial separations.

You can even take it up a level and say that interaction events are the only things that really exist. That's more or less what Copenhagen QM boils down to. It is not at all a given that "particles" even exist between interactions.

Intuition insists that if something happens here and something apparently related happens over there and you can move here and there around to make a line or curve, then something physical is moving between here and there.

But actually - no. Not necessarily. All you have are ghost traces of an apparent chain of causality. And you can play with those traces experimentally to make them do incredibly weird shit in very surprising ways.