Just a partial answer, for dimension 2 and 3, the answer is yes, there are infinitely many cocompact hyperbolic Coxeter groups. Furthermore, in these dimensions the commensurability classes of the groups can be distinguished by their respective invariant trace fields.

For $n=3$, it follows from $\S 4.7.3$ of Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds." In this section, the explicitly construct infinitely many distinct hyperbolic Coxeter groups generated by reflections in the faces of a compact prism. Then they have as an exercise (4.7.4 on page 151) that the invariant trace fields of their examples only depend on a free parameter. In particular, the invariant trace fields can be arbitrarily high degree extensions of $\mathbb{Q}$.

For $n=2$, one could use hyperbolic triangle groups. To set notation, the groups can be distinguished by their orientation preserving index 2 subgroups of the form $\langle x,y | x^m=y^n=(xy)^p=1 \rangle$. For a fixed $(m,n,p)$, the invariant trace field is $\mathbb{Q}(\cos \frac{2\pi}{m},cos \frac{2\pi}{n}, \cos \frac{2\pi}{p},\cos \frac{\pi}{m} \cos \frac{\pi}{n} cos \frac{\pi}{p})$. Again, if the triple $(m,n,p)$ is varied, this field can be made an arbitrarily high degree extension of $\mathbb{Q}$.

Just a partial answer, for dimension 2 and 3, the answer is yes, there are infinitely many cocompact hyperbolic Coxeter groups. Furthermore, in these dimensions the commensurability classes of the groups can be distinguished by their respective invariant trace fields.

For $n=3$, it follows from $\S 4.7.3$ of Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds." In this section, the authors explicitly construct infinitely many distinct hyperbolic Coxeter groups generated by reflections in the faces of a compact prism. Then they have as an exercise (4.7.4 on page 151) that the invariant trace fields of their examples only depend on a free parameter. In particular, the invariant trace fields can be arbitrarily high degree extensions of $\mathbb{Q}$.

For $n=2$, one could use hyperbolic triangle groups. To set notation, the groups can be distinguished by their orientation preserving index 2 subgroups of the form $\langle x,y | x^m=y^n=(xy)^p=1 \rangle$. For a fixed $(m,n,p)$, the invariant trace field is $\mathbb{Q}(\cos \frac{2\pi}{m},cos \frac{2\pi}{n}, \cos \frac{2\pi}{p},\cos \frac{\pi}{m} \cos \frac{\pi}{n} cos \frac{\pi}{p})$. Again, if the triple $(m,n,p)$ is varied, this field can be made an arbitrarily high degree extension of $\mathbb{Q}$.

Just a partial answer, for dimension 2 and 3, the answer is yes, there are infinitely many cocompact hyperbolic Coxeter groups. Furthermore, in these dimensions the commensurability classes of the groups can be distinguished by their respective invariant trace fields.

For $n=3$, it follows from $\S 4.7.3$ of Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds." In this section, the explicitly construct infinitely many distinct hyperbolic Coxeter groups generated by reflections in the faces of a compact prism. Then they have as an exercise (4.7.4 on page 151) that the invariant trace fields of their examples only depend on a free parameter. In particular, the invariant trace fields can be arbitrarily high degree extensions of $\mathbb{Q}$.

For $n=2$, one could use hyperbolic triangle groups. To set notation, the groups can be distinguished by their orientation preserving index 2 subgroups of the form $\langle x,y | x^m=y^n=(xy)^p=1 \rangle$. For a fixed $(m,n,p)$, the invariant trace field is $\mathbb{Q}(\cos \frac{2\pi}{m},cos \frac{2\pi}{n}, \cos \frac{2\pi}{p},\cos \frac{\pi}{m} \cos \frac{\pi}{n} cos \frac{\pi}{p})$. Again, for as $(m,n,p)$ is varied, this field can be made an arbitrarily high degree extension of $\mathbb{Q}$.

Just a partial answer, for dimension 2 and 3, the answer is yes, there are infinitely many cocompact hyperbolic Coxeter groups. Furthermore, in these dimensions the commensurability classes of the groups can be distinguished by their respective invariant trace fields.

For $n=3$, it follows from $\S 4.7.3$ of Maclachlan and Reid's "The Arithmetic of Hyperbolic 3-manifolds." In this section, the explicitly construct infinitely many distinct hyperbolic Coxeter groups generated by reflections in the faces of a compact prism. Then they have as an exercise (4.7.4 on page 151) that the invariant trace fields of their examples only depend on a free parameter. In particular, the invariant trace fields can be arbitrarily high degree extensions of $\mathbb{Q}$.

For $n=2$, one could use hyperbolic triangle groups. To set notation, the groups can be distinguished by their orientation preserving index 2 subgroups of the form $\langle x,y | x^m=y^n=(xy)^p=1 \rangle$. For a fixed $(m,n,p)$, the invariant trace field is $\mathbb{Q}(\cos \frac{2\pi}{m},cos \frac{2\pi}{n}, \cos \frac{2\pi}{p},\cos \frac{\pi}{m} \cos \frac{\pi}{n} cos \frac{\pi}{p})$. Again, if the triple $(m,n,p)$ is varied, this field can be made an arbitrarily high degree extension of $\mathbb{Q}$.