Your claim that every discrete group of isometries of $\mathbb E^2$ is equal to a finite index sugroup of a Coxeter group is not true. One can infer this from a dimension count, as follows.
Let's consider the space of discrete groups $\Gamma$ of isometries of $\mathbb E^2$ with the following properties:
- $\Gamma$ is isomorphic to $\mathbb Z \times \mathbb Z$,
- $\Gamma$ is normalized (up to conjugation by a similarity) to contain the translation $T_{(1,0)}$, meaning that $T_{(1,0)}(x,0)=(x+1,y)$, and so that $T$ has minimal translation length amongst all elements of $\Gamma$.
This space is 2-dimensional, as you can see by noticing that there exists an element $T_{a,b}$ in the group (meaning that $T_{(a,b)}(x,y)=(x+a,y+b)$) such that $(a,b)$ is contained in the region defined by the inequalities $a^2+b^2 \ge 1$ and $0 \le a \le 1$, and furthermore this element is almost unique: $\Gamma$ contains at most 2 such elements $T_{(a,b)}$, and 2 of them occur only if $(a,b)$ lies on the boundary of the region, meaning that $a^2+b^2=1$ or $a=0$ or $a=1$.
The Coxeter triangle groups are all rigid up to normalization, and so their finite index subgroups form a countable subset of the space of discrete groups.
The only Coxeter groups that are not triangle groups are the rectangle reflection groups, and they form just a 1-dimensional space. Their finite index subgroups therefore form a 1-dimensional subset of the space of discrete groups, namely those for which there exists some element of the form $T_{(0,b)|$, $b \ge 1$. Your up/down, left/right example in your comment does lie in this subspace.
Using this analysis, for an explicit example that is not a finite index subgroup of a Coxeter group, take the group generated by $T_{(1,0)}$, $T_{(1/2,1)}$.
A similar dimension count applies to discrete groups of isometries of $\mathbb H^2$. This gets into the topic of hyperbolic surfaces and hyperbolic 2-orbifolds and their Teichmuller spaces. I'll give just the simplest example (which is already too complicated to describe in full detail, so I'll have to leave a lot of claims unproved). Consider the abstract group $\pi_1(S_2)$ which is the fundamental group of the closed oriented surface $S_2$ of genus $2$, defined by the presentation
$$\pi_1(S_2) = \langle a,b,c,d \mid aba^{-1}b^{-1}cdc^{-1}d^{-1} \rangle
$$
This group has a Teichmuller space, which is the space of (suitably normalized) discrete subgroups of isometries of $\mathbb H^2$ that are isomorphic to $\pi_1(S_2)$. One can do a dimension count to show that this space is 6-dimensional. Also, the Euler characteristic of $S_g$ is equal to $\chi(S_g)=2-2g$ in general, and $\chi(S_2)=-2$.
If $\pi_1(S_2)$ is isomorphic to a finite index subgroup of a Coxeter group, then the corresponding reflection polygon $P$ has a rational Euler characteristic $\chi(P)$ which is a rational number strictly between $-2$ and $0$. These reflection polygons can be explicitly enumerated, and the dimensions of their Teichmuller spaces all calculated explicitly, and one sees that in each case the dimension is strictly less than $6$.
Best Answer
Consider the following result, which appears as Proposition 6.8 in Humphreys (with further comments on p 140), and Exercise 14 in chapter V of Bourbaki.
Here compact hyperbolic means that the standard geometric representation $(W,S)$ on $\mathbb{R}^n$ induces an action on hyperbolic space modelled as one component of $\{\lambda\in \mathbb{R}^n\mid B(\lambda,\lambda)=-1\}$ which is discrete and cocompact.
Now is an appropriate point to mention that the (compact) hyperbolic groups have indeed been classified, and that classification can be found in $\S$6.9 of Humphreys or exercise 15 of Bourbaki. I have also included them all here, note there are finitely many, and only in 3, 4, and 5 dimensions.
If you are happy for your special subgroups to be positive semi-definite, rather than positive definite, then replace compact hyperbolic in the above theorem with hyperbolic (cf Humphreys Proposition 6.8 and exercise 13 in Bourbaki).
Clearly as you say, if $(W,S)$ is finite and irreducible, then all its special subgroups are also finite, so the only remaining case to consider is when $B$ is degenerate. For this case we can appeal to Proposition V.4.10 in Bourbaki which essentially says the following
The classification of these irreducible affine Coxeter groups is well-known and each satisfies the condition you want on special subgroups.
The only case not covered is if $B$ is degenerate and not positive. I am not familiar with any results on this class of Coxeter groups I am afraid. You may be reduced to considering all possible extensions of finite type Coxeter diagrams. This probably wouldn't be quite as bad as it first seems because the proof of the classification of finite Coxeter groups gives very tight conditions on the local structure of their diagrams which would imply something like the following:
And possibly a few others, so actually the possibilities are very limited.