Derivation of Transfer Functions for Kalman Filter
This is an attempt to answer. It's not complete, and a little fuzzy, but feel free to comment and I can edit / update.


  1) the statement about the determinant of $(I-Fz)$ being 1 when $z$ = zero.
  I mean I understand why that's true but what does it have to do with the price of tea in china ? or causality ?


So let's have a look:
$$
x(t+1) = Fx(t) + Lz(t) + K \epsilon(t)\\
zx(t) - Fx(t) = Lz(t) + K \epsilon(t)\\
(I-Fz^{-1})x(t) = z^{-1}[Lz(t) + K \epsilon(t)]\\
x(t) = (I-Fz^{-1})^{-1}z^{-1}[Lz(t) + K \epsilon(t)]\\
$$

One thing to note about this is that $z$ is used twice: once as the lag operator and once as the "reference" signal.  That confused me for a while. Another thing is that they appear to use $z$ as the lag operator, whereas $z^{-1}$ is usually used. Above I've assumed the conventional approach to $z$ as the advance operator.

My understanding of causality comes from the ROC associated with the $z$ transform.  A $z$-transform is causal if its ROC extends outwards from the outermost pole.  A $z$-transform is BIBO stable if its ROC includes the unit circle.

I believe what they really should have said is:


  Because we assume that $\displaystyle(I - Fz)^{-1} = \sum_{j = 0}^\infty F^{j}z^{j}$, we assume $j\not &lt; 0$ and so assume causality.


which is also sort of implied in their use of state space equations, as they say In this sense, causality is built into the definition of state-space systems.


  2) the derivation of the transfer functions, $k(z)$ and $l(z)$. I do understand 1.2.21) so I imagine that 1.2.21) is used to derive them ????? I also understand the second equalities in $k(z)$ and $l(z)$. But not the first equality signs.


Assuming for now that $L=0$, then the state space form of the $z$ transform of (1.1.22) and (1.1.23) is just:
$$
H(I - z^{-1})K + I
$$
see these notes, particularly equation (10.20):



Though they use $A, B, C,$ and $D$ instead of $F, K, H,$ and $I$.

This is just the standard way to represent transfer functions in state space.


  3) the equation for $x(t)$ at the very end. I do see how $y(t)$ is obtained once one has $x(t)$.


I believe the $x(t)$ equation is just had by the usual solution to a linear, constant coefficient difference equation with non-zero initial conditions (but using their notation). It's not the same notation, but try this.


  My guess is that I need is nice reference or lectures notes that explain why, when it comes to the KF, 
  
  A) the determinant of $(I-Fz)$  is important and 


See answer to 2) above.


  B) how the associated transfer response functions are obtained  from a state space representation? 


See references above.


  C) How the $x(t)$ equation is obtained given the transfer function ? 


See answer to 3) above.