The answer to your basic question is yes, though the notion of "Weyl chamber" comes up in somewhat different contexts (so it has to be defined carefully). Abstractly, the Bourbaki development of "root systems" leads to a simple geometric picture in a real vector space: you have hyperplanes orthogonal to finitely many pairs of roots, and the resuting Weyl chambers are defined to be the (open) connected components in the complement of the union of hyperplanes. In turn, the Weyl group generated by reflections in the hyperplanes will act simply transitively on the Weyl chambers, which are in particular pairwise disjoint.

Concretely, all of this arises in the traditional theory of Lie groups, most transparently in the extreme cases where the group is compact semisimple and where it is "split"; so its complexification is one of the usual semisimple groups over $\mathbb{C}$ and can also be regarded as a semisimple algebraic group. In general, for a real semisimple Lie group, you get a relative root system and relative Weyl group, along with its "Weyl chambers". But in any case the Weyl chambers are mutually disjoint.

There are quite a few sources for the standard Lie group structure theory. Look for example at the modern book by A.W. Knapp, Representation Theory of Semisimple Lie Groups. Here Chapter 3 works out the structure theory for noncompact groups, and in section 3 he treats the Weyl group and Weyl chambers in a notational framework similar to yours.

The answer to your basic question is yes, though the notion of "Weyl chamber" comes up in somewhat different contexts (so it has to be defined carefully). Abstractly, the Bourbaki development of "root systems" leads to a simple geometric picture in a real vector space: you have hyperplanes orthogonal to finitely many pairs of roots, and the resulting Weyl chambers are defined to be the (open) connected components in the complement of the union of hyperplanes. In turn, the Weyl group generated by reflections in the hyperplanes will act simply transitively on the Weyl chambers, which are in particular pairwise disjoint.

Concretely, all of this arises in the traditional theory of Lie groups, most transparently in the extreme cases where the group is compact semisimple and where it is "split"; so its complexification is one of the usual semisimple groups over $\mathbb{C}$ and can also be regarded as a semisimple algebraic group. In general, for a real semisimple Lie group, you get a relative root system and relative Weyl group, along with its "Weyl chambers". But in any case the Weyl chambers are mutually disjoint.

There are quite a few sources for the standard Lie group structure theory. Look for example at the modern book by A.W. Knapp, Representation Theory of Semisimple Lie Groups. Here Chapter 3 works out the structure theory for noncompact groups, and in section 3 he treats the Weyl group and Weyl chambers in a notational framework similar to yours.