It's simpler than that. A beat frequency occurs whenever two waves of slightly different frequency add together.
Again, it's much simpler than you make out. Two events are simultaneous in a particular frame of reference if their time coordinates are the same. That's all there is to it.
I can set up a coordinate system right here in my room. I might choose, for example, to designate the origin of my coordinate system as the front left-hand corner of my desk. If I want to, I could add an origin for the time coordinate to the definition (e.g. time zero occurs at the front left-hand corner of my desk at 8 pm on 25 December, 2016).
I don't think your idea of "inertialess relativistic space" is required or important to defining a coordinate system.
Oh, and my desk-based coordinate system can be agreed upon by as many observers in as many frames of reference as you like.
The task of relativity is to translate my (arbitrary) coordinates into any other set of coordinates that any other observer would care to set up.
Relativity is about moving from one reference frame to another. If by "observer" you mean "reference frame", then you are correct. We don't need relativity until we have more than one reference frame.
I don't know what "single mathematical expression" you might be thinking of.
No. Simultaneity is defined in a single reference frame. In fact, I gave the definition earlier in this post.
Time dilation of what?, I wonder...
Time dilation of one region of inertialess, relativistic, non-Euclidean space 'light travel time' relative to any other measure of the rates of energy transfer events in a different time dilation zone of the same, or of a different bound or unbound energy wave of bulk energy propagation, of course. Imagine what we call 'space' as beads of rotationally bound energy separated from each other by linear light travel time. No other description of space is really needed then, is it? The speed of light in a vacuum adequately describes energy transfer events in both (rotational and linear) domains, and the continuous interactions between the two are determined by the substrate quantum field. This would include entanglement, the arrow of time, the speed of light for bulk boun or unbound energy propagation, and its relationship to 'at rest' for bound energy. And any fixed measure of 'space' or coordinate systems not related to simple light travel time would be a ludicrous waste of mathematical effort.
Not so simple, trying to be consistent in the description of events in such a universe. But Minkowski's description was far worse. For people like him, his precious quadratics and faux simultanaeity were only a mathematical refuge from struggling with descriptions of physical reality, the way his lesser student Einstein did. Hilbert was always Minkowski's favorite son.
I have so far only defined energy in mathematical terms, using Einstein and Planck's relations. Kinetic energy is an excitation of a quantum field. Potential energy is more complex and always, always involves bound energy, which makes the situation considerably more complex in terms of descriptive power.
As for your example of a coordinate system in a room, every atom of every solid, liquid, gas, or plasma in that room is riding on a planet, moon, star, or spacecraft moving in terms of energy exchange events in an inertialess, non-Euclidean quantum field which is isotopic in every direction outside the influence of a gravitational field, which is another type of energy transfer event based on quantum spin of that field. The atomic and fundamental particle structures of everything in that room including yourself are dynamic; not static, and not Euclideann. There are places where your coordinate system is appropriate to apply and places where it is not. But it will definitely not serve very well to describe or to understand either the physical natures nor mechanisms underlying inertia nor gravity.
The OP hinges on SFA.
