1)Definition: An inertial frame of reference is a frame of reference where Newton's first law applies (uniform motion if without external force).
Now if we have other frame of references that are moving relative to this inertial frame with
uniform relative velocities, then all the others are also called inertial frame of references.
2)Transformation between inertial reference frames:In Newtonian mechanics, the laws of physics are invariant under Galilean transformation. While in special relativity, the laws of physics are invariant under Lorentz transformation. The latter reduces to the former in classical limit.
The basic assumptions on the space-time structure in classical mechanics are:
(1) Time intervals beetween events are absolute.
(2) Space intervals beetween contemporary events are absolute.
We may refere to this two properties as to the "galilean space-time structure". In the first chapter of Arnold's "Mathematical Methods of Classical Mechanics" you can find a mathematical formulation of space-time structure, and one of the problems is to prove that
All the affine transformations of space time which preserve time intervals and distances beetween contemporary events are compositions of rotations, translations and uniform motions.
The principle of relativity alone allows a much wider class of transformations (for example, it allows Lorentz's transformations).
So I think that your second and third questions can be rephrased in this way:
Can the galilean structure of space-time be obtained from known equations of motions (and the principle of relativity)?
I'm not able to be cathegorical about this point, but my intuition says that the answer is no, without at least some other strong assumptions.
Here's a simple derivation of galilean transformations.
Let $(t,\mathbf r)$, $(t',\mathbf r'$), denote the coordinates of an event in two frames of reference (not necessarily inertial frames).
In first place, the invariance of time intervals beetween events implies that
$$t'=t+t_0,$$
for some constant $t_0$.
Now, the invariance of space intervals beetween simultaneous events implies that, for a fixed $t$, $\mathbf r \mapsto \mathbf r'$ is an isometry of $\mathbb R ^3$. The most general form of such isometry is: $$\mathbf r'= \mathbf s+G\mathbf r,$$
where $GG^T=I$ and both $\mathbf s$ and $G$ may depend on time.
To establish that $\dot {\mathbf s}=0$ and $\dot G = 0$ if the two frames are inertial we notice that, by the principle of relativity, the equation: $$\ddot {\mathbf r}=0$$ for an isolated body must be covariant; a direct calculation shows that this is possible only if the above conditions are satisfied.
Best Answer
Maybe just to emphasize the main point of Andrew steane's answer: The crucial connection is that charged particles are both mechnaical and electrodynamical objects.
Assume that you have some reference frame where you consider a bunch of charged particles. They will move according to ther mechanical equations of motions under the mutual forces determined by the electromagnetic fields, which are, in turn, generated by the charge particles themselves.
If mechanical and electrodynamical would change differently under changes of reference frame, this picture would change, and either the mechanical or the electrodynamical laws (or both) would have to change.