The answer for question (0) about $U_{14}$ is positive, as described in the Conway-Odlyzko-Sloane paper:
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 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 $Aut(U_{14})=W(E_7) \wr C_2$.
EDIT: The answer above also solves (3B) for $U_{14}$ and $A_{15}^+$: for $U_{14}$, the closest vectors are the union of two orthogonal $2_{31}$ polytopes; and for $A_{15}^+$, the closest vectors are those vectors in $\mathbb{R}^{16}$ containing a $1$, a $-1$ and 14 $0$s.