- The answer for question (0) about $U_{14}$ is positive, as described in the [Conway-Odlyzko-Sloane paper](https://www.cambridge.org/core/journals/mathematika/article/abs/extremal-selfdual-lattices-exist-only-in-dimensions-1-to-8-12-14-15-23-and-24/A525C50AC72B084E4E453E1159A9BCF7): 

  Let $E_8$ be the lattice in $\mathbb R^8$ with its eight co-ordinates all in $\mathbb Z$, or all in $\mathbb Z + 1/2$, with even sum and $E_7$ be the sub-lattice of $E_8$ with $x_1 + ... + x_8 = 0$.

  Then $U_{14}$ is the sum of $E_7+E_7$ and the vector $((\frac14)^6,(-\frac34)^2,(\frac14)^6,(-\frac34)^2)$.

- The answers for question (0) about $Q_{14}$, $A_{15}^+$, question (1) and question (2) may be negative, as indicated by the vast differences of their automorphism groups:

  For each lattice, I computed the vectors closest to $0$ and study the automorphism group (as a subgroup of orthogonal group) of those vectors. This is a supergroup of the true automorphism group.

  The results are:

  | Group | Aut(closest vectors)|
  |---------|-------------|
  |  $U_{14}$       |    $W(E_7)  \wr C_2$            |              
  |     $Q_{14}$    |   $C_2 \times G_2(3)$                |                
  |      $A_{15}^+$   |      $C_2 \times S_{16}$             |   

  where $W(E_7)$ is the Weyl group of $E_7$ and $\wr$ is the wreath product.

  The automorphism group of $Q_{14}$ is [already computed](https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/P14.1.html) and it is  $C_2 \times G_2(3)$.

  The definitions given in the Conway-Odlyzko-Sloane paper indicates that its automorphism group contains $S_{16}$, so we have $S_{16} \subset Aut(A_{15}^+) \subset C_2 \times S_{16}$.

  For $U_{14}$, I have also computed the automorphism group of vectors with distance $\sqrt 2$ and $\sqrt 3$ from $0$, and it's also $W(E_7)  \wr C_2$. As the vectors with distance $\sqrt 2$ and $\sqrt 3$ from $0$ generate the whole $U_{14}$, it turns out that $Aut(U_{14})=W(E_7)  \wr C_2$.