Flipping-Cups Vector Sequences

For an $n$-dimensional vector $A=(a_1,\dots,a_n)$ with each $a_i\in\{0,1\}$, call $A$ an $n$-dimensional T-vector. For two such vectors $A,B$, define $d(A,B)=\sum_{i=1}^n|a_i-b_i|$.

1. For $A=(1,0,1,0,1)$ and $B=(0,1,1,1,0)$, find $d(A,B)$. 2. Consider a sequence of $5$-dimensional T-vectors $A_1,A_2,\dots$ with $A_1=(1,1,1,1,1)$ and $d(A_i,A_{i+1})=2$ for all $i$. Prove that the zero vector $(0,0,0,0,0)$ never appears in the sequence. 3. Consider a sequence of $12$-dimensional T-vectors $A_1,A_2,\dots$ with $A_1=(1,1,\dots,1)$ and, for some $j\in\mathbb N^{*}$, $A_j=(0,0,\dots,0)$; suppose there is a positive integer $m$ with $d(A_i,A_{i+1})=m$ for all $i$. Find all possible values of $m$.