The following is essentially a contribution to point 2.
Non-commutative (which can be specialized to matrices or scalars)
continued fraction are used in enumeration and language theories.
Two examples : Dyck and Motzkin paths.

These are two lattice paths
drawn on $\mathbb{N}^2$ (the first quarter-plane).

**Dyck Paths** Steps are $a=(1,1)$ (north-east step) and $b=(1,-1)$
(south-east step) Dyck paths are defined as paths that

- start at $(0,0)$
- end at $(0,2n)$ (for some $n$)
- always stand above the $x$-axis

They can be coded by words $w$
(Dyck words)
in the alphabet $\{a,b\}$ such that, if
$w=uv$ one has $|u|_a\geq |u|_b$ (the path is always in $\mathbb{N}^2$) and,
at the end $|w|_a=|w|_b$ (it returns to the $x$-axis). In what preceeds
$|u|_a$ (resp. $|u|_b$) stand for the number of occurences of $a$ (resp. $b$)
in the word $u$.

Let $D$ be the set of Dyck words. It can be shown that it is a free monoid
with alphabet (irreducible Dyck) $A$ (the paths which return to the $x$-axis
only at the end). Indentifying them with their characteristic series
$$
D=\sum_{w\atop\small{Dyck\ word}}\,w\ ;
A=\sum_{w\atop\small{Irreducible\ Dyck\ word}}\,w
$$
(computed in $\mathbb{Z}\langle\langle a,b\rangle\rangle$) one has $A=aDb$
and $D=(1-A)^{-1}$, then

$$
A=a(1-A)^{-1}b=a\frac{1}{1-A}b=a\frac{1}{1-a\frac{1}{1-A}b}b=a\frac{1}{1-a\frac{1}{1-a\frac{1}{1-A}b}b}b=\ldots
$$
with suitable structures, one can show that this continued fraction converges.

**Motzkin Paths** Similarly, but with steps are $a=(1,1)$ (north-east step), $c=(1,0)$
(east step, i.e. constant) $b=(1,-1)$ (south-east step)
Motzkin paths are defined as paths that

- start at $(0,0)$
- end at $(0,n)$ (for some $n$)
- always stand above the $x$-axis

They are coded by Motzkin words which form again a free monoid $M$ with alphabet, say $B$
of (irreducible Motzkin words). One has $B=(c+aMb)$ and $M=(1-B)^{-1}$ thus
$$
B=c+aMb=c+a\frac{1}{(1-B)}b=\ldots
$$
Hope this helps. Do not hesitate to interact.