Is there a naming convention for the left child of a right child (and vice versa)?
You can use 'sibling' (or 'brother'/'sister' if you prefer to gender the vertices). It's somewhat commonly used, but you may want to redefine it quickly when using it extensively.

NB: this terminology can be used for non-binary tree, the sibling of a vertex is any other vertex with the same parent/father.

While you may want to find a more reliable source, Wikipedia agrees with me at the moment: https://en.m.wikipedia.org/wiki/Tree_structure