Have you looked at small values of $m$? If a pattern shows itself, that would be encouraging. Nothing about the questions suggests to me it will have a nice answer, but I haven't really got much intuition for the question.

The parabolic decomposition $G=\sqcup_{w\in W^P} BwP$ suggests that the parabolic subgroups of $G$ over $\mathbb F_1$ are also the parabolic subgroup $W_P$.

Folding always comes from a non-trivial group of Dynkin diagram automorphisms, which can be thought of as having an action on the same vector space on which $W$ acts as a reflection group. It seems reasonable to guess that the folded group is the subgroup of $W$ commuting with this action. Maybe it should be obvious but I don't immediately see it.

@PerAlexandersson: You mean 5 isosceles triangles. The fact that your interpretation makes the area well-defined (which it otherwise isn't) makes me think your interpretation of "cyclic" is the right one.