How do you know?

The short answer is that energy eigenstates don't change over time, while the universe does.

If you add the waveform of a system where particle a has energy 1 and particle b has energy 2 to a system where particle a has energy 2 and particle b has energy 1, you end up with a system with an energy eigenstate of 3, but each particle is not in an energy eigenstate.

This is a good point. What I said didn't mean what I thought it meant. But this system seems like an example of the power of entanglement. If the particles were unentangled, there would be change over time. But they are entangled, and there isn't any change over time. A computer living in this system would not actually move any electrons around to do any computation.

What is an "event"?

An event is a "thing that happens". Relativity made discussion of "events" routine in physics, because one wants to talk about something - the tick of a clock, the emission or absorption of a photon - that is localized in space and time. "Event" is a completely standard term of art in relativity - thus "event horizon". Of course, it is also an elementary everyday word and concept, independent of its use in physics.

In standard probability calculus, P((A and B) or (C and D)) = P(A) x P(B) + P(C) x P(D). You're summing the probabilities of two possibilities: the possibility that A and B occur together, and the possibility that C and D occur together. Feynman's reformulation of quantum mechanics as a "sum over histories" has this schematic form as well, except that it is the complex-valued "probability amplitudes" that we multiply and add. The basic events are "a particle moves from one place to another place" and "a particle is emitted or absorbed", and the amplitude or these events is "e" to the power of "i" times the "action" for this event, action being a concept from classical physics which carries over to quantum theory, and which in fact assumes a fundamental role there.

A standard example of a timeless-looking construction from quantum cosmology is the Hartle-Hawking wave-function of the universe, derived from a "no-boundary condition". This wavefunction assigns an amplitude of three-dimensional configurations of the universe; which sounds like Barbour's "Platonia", But how are these amplitudes calculated? By summing over space-time histories which evolve to the three-dimensional configuration of interest. The amplitude for a configuration X is the sum of the amplitudes of every space-time history which starts from "nothing" (that's why this is the "no-boundary condition") and which evolves to X.

In a timeless framework, you could possibly conceive of an event as the difference between one configuration and its neighbor in configuration space, and the amplitude for the "event" as the weighting for the contribution made by the first configuration to the amplitude of the second configuration, via timeless amplitude "flow". That is, if you have one configuration of particles, and then a neighboring configuration which is the same except that there is now an extra photon on top of one of the electrons, then the "event" corresponding to this configurational difference would be "emission of a photon by that electron", and the usual Feynman amplitude for this event would define the proportional contribution to the amplitude flow entering timelessly into the second configuration's point in configuration space.

It's a standard fact about the Schrodinger and Feynman formulations of quantum mechanics that they are equivalent - the evolution of the Schrodinger wavefunction is equivalent to the cumulative flow of amplitude produced by the converging and diverging Feynman histories - and this should carry over to the timeless case of quantum cosmology... in principle. But in practice, the Feynman formulation seems more relativistic so perhaps it's more fundamental. In any case, you do sometimes multiply amplitudes when you do quantum mechanics as Feynman did it.