These may be "negative results" for the first three questions except the U14 branch of the zeroth question.
The one-point stabilizer of U14 has structure $(O_7(2) \times O_7(2)) : D_8$ (the colon represents a split extension).
The one-point stabilizer of C2G23 has structure $C_2 \times G_2(3)$.
The one-point stabilizer of A15+ has structure $C_2 \times S_{16}$.
These are computed by finding the vectors closest to $0$ of the respective lattices, and study the action of the stabilizer of $0$ on the closest vectors. The rest is just standard GAP permutation group calculation, e.g. using the StructureDescription command.