Starting at the top and working my way down.
Good questions, by the way!
So basically state and output is modelled as gaussian distributions
that have slowly changing means (to be estimated), with additive
zero-mean gaussian noise on top (to be cancelled)?
Yes, that's correct. The KF formulation can work with other distributions, but the standard ...
Since for a transmission line, the system is distributed since RLC is distributed in infinitesimally small sections in the line. The t values can be assumed to be infinitesimally small sections of the unit time delay system.
However, for me, a distributed system is something like a transmission line, where the parameters can't be modeled accurately by "discrete" and finite elements.
I can't grasp why this system is classified as distributed just because there are an infinite number of points describing the initial state.
It's not just that you need an infinite number of ...
You are right that a distributed system could be "something like a transmission line". Note that the system
is a simple model of a transmission line, where just a frequency-independent delay $T$ is taken into account, and the attenuation is neglected.
Note that lumped electrical systems, described by resistors, capacitors and ...