For an even integral lattice $L$ (of arbitrary signature, non-degenerate but not necessarily unimodular), consider roots (vectors of square $2$) and their orthogonal hyperplanes.

First, I'd like to know if the collection of root hyperplanes is always locally finite away from the origin. (I'm not actually sure what this means, but I have come across this condition in many papers. At the very least, it should mean that every non-zero point in $L \otimes \mathbb{R}$ has a neighborhood meeting a finite number of the hyperplanes.)

Second, if the above is true, I can take the complement of all of these hyperplanes to be left with a bunch of connected components and I'd like to know if choosing one of them always gives a fundamental domain for the group generated by reflections in the root hyperplanes. I know that this is the case for finite and affine root systems, and am wondering if this is a general fact.