A Jump of Size $m$

Let $A \colon a_1, a_2, \ldots, a_{2m}$ be a permutation of $1, 2, \ldots, 2m$, where $m \in \mathbb{N}^*$ and $m \geqslant 3$. Say $A$ has property $P$ if there is at least one $i \in \{1, 2, \ldots, 2m-1\}$ with $|a_i - a_{i+1}| = m$.

1. For $m = 3$, decide whether the sequences $B \colon 1, 5, 3, 4, 6, 2$ and $C \colon 6, 5, 2, 4, 1, 3$ have property $P$. 2. Suppose the subsequences $\{a_{2n-1}\}$ and $\{a_{2n}\}$ ($n = 1, 2, \ldots, m$) are both arithmetic, with $a_1 = 1$ and $a_{2m} = 2$. Prove that for every even $m$, the sequence $A$ does not have property $P$. 3. Among all permutations of $1, 2, \ldots, 2m$, let $S$ be the number of sequences with property $P$ and $T$ the number without it. Prove that $S > T$ for every $m \geqslant 3$.