= Solution
Put $B=\partial M$ and let $v$ be the cone vertex. Specify the convention away from the Witt case: first take <intersection cohomology> to mean the <hypercohomology> of the <intersection complex> $IC_{\bar m}$, normalized in degree zero on the regular stratum. The <lower middle perversity> at the vertex is $\bar m(2m+1)=m-1$. The standard local cone calculation for this convention says
$$
IH_{\bar m}^i(\widehat U)=\begin{cases}H^i(B;\mathbb Q),&i<m,\\0,&i\ge m,\end{cases}
$$
and restriction to $U\simeq B$ is the identity below the cutoff. Here this can also be read directly from the construction $IC_{\bar m}=\tau_{\le m-1}Rj_*\mathbb Q$, with $j:\widehat M\setminus\{v\}\hookrightarrow\widehat M$: punctured conical neighbourhoods retract onto $B$, and the truncation discards the degrees above $m-1$.
Define relative <intersection cohomology> as the <hypercohomology> of the homotopy fibre of restriction, namely $R\Gamma(\widehat M,\widehat U;IC)=\operatorname{Cone}(R\Gamma(\widehat M;IC)\to R\Gamma(\widehat U;IC))[-1]$. Restricting to the pair $(M,U)$ gives a map to the corresponding ordinary relative <cohomology>, since $IC|_M=\mathbb Q_M$. The induced map of <exact triangles> gives the requested <commutative diagram>:
$$
\begin{array}{ccccccccc}
\cdots\to&IH^i(\widehat M,\widehat U)&\to&IH^i(\widehat M)&\to&IH^i(\widehat U)&\to&IH^{i+1}(\widehat M,\widehat U)&\to\cdots\\
&\downarrow&&\downarrow&&\downarrow&&\downarrow&\\
\cdots\to&H^i(M,U)&\to&H^i(M)&\to&H^i(U)&\to&H^{i+1}(M,U)&\to\cdots .
\end{array}
$$
Both rows are <long exact cohomology sequences of a pair>. Commutativity includes the <connecting homomorphisms>, because they arise from the same map of restriction cones, with a common sign convention. The relative vertical maps are isomorphisms: <excision> removes the common conical neighbourhood, leaving precisely the <manifold> pair outside it, where $IC$ is constant. Also the collar retracts onto $B$, so $H^*(M,U)\cong H^*(M,B)$.
The only possible nonmanifold stratum is $v$, with link $B$ of dimension $2m$. Since <intersection homology> of a <manifold> is ordinary homology, the exact necessary and sufficient condition is
$$
\boxed{\widehat M\text{ is Witt}\quad\Longleftrightarrow\quad H^m(B;\mathbb Q)=0.}
$$
The equivalence with the defining $H_m(B;\mathbb Q)=0$ follows from the <universal coefficient theorem for cohomology> over $\mathbb Q$. In the Witt case the cone restriction is an isomorphism through degree $m$, since both degree-$m$ groups are zero. Comparing the two exact rows, or applying the <Five lemma> to each five-term segment, gives $IH^i(\widehat M)\cong H^i(M)$ for $i\le m$. For $i>m$, both $IH^{i-1}(\widehat U)$ and $IH^i(\widehat U)$ vanish, so the top row identifies the absolute group with the relative one. Hence
$$
\boxed{IH^i(\widehat M)\cong\begin{cases}H^i(M;\mathbb Q),&i\le m,\\H^i(M,B;\mathbb Q),&i>m.\end{cases}}
$$
Without the Witt hypothesis, comparison still gives $IH_{\bar m}^i(\widehat M)\cong H^i(M)$ for $i<m$. At degree $m$ the upper row reads
$$
H^{m-1}(B)\xrightarrow{\delta}H^m(M,B)\longrightarrow IH_{\bar m}^m(\widehat M)\longrightarrow0.
$$
The lower row identifies the quotient by $\operatorname{im}\delta$ with the image of the relative-to-absolute map. In higher degrees both cone terms vanish. Thus the full lower-middle Deligne answer is
$$
\boxed{IH_{\bar m}^i(\widehat M)\cong\begin{cases}H^i(M),&i<m,\\\operatorname{im}(H^m(M,B)\to H^m(M)),&i=m,\\H^i(M,B),&i>m.\end{cases}}
$$
For clarity, if $IH^i$ is instead defined as the linear dual of the lower-middle <intersection chains> of question 1, it corresponds to the complementary <upper middle perversity> Deligne <sheaf>. Its cone group retains $H^m(B)$ as well. The same diagram then gives
$$
\boxed{IH_{\bar n}^i(\widehat M)\cong\begin{cases}H^i(M),&i\le m,\\\operatorname{im}(H^{m+1}(M,B)\to H^{m+1}(M)),&i=m+1,\\H^i(M,B),&i>m+1.\end{cases}}
$$
Equivalently the exceptional group is $H^{m+1}(M,B)/\operatorname{im}(H^m(B)\xrightarrow{\delta}H^{m+1}(M,B))$. These conventions agree under the stated Witt condition, explaining why that condition removes the ambiguity.
Take the solid <torus> $M=S^1\times D^2$, whose boundary is the two-dimensional <torus> $B=T^2$. Its ordinary <cohomology> is $H^0(M)=H^1(M)=\mathbb Q$, zero otherwise; <Poincare duality> for a <manifold> with boundary gives $H^2(M,B)=H^3(M,B)=\mathbb Q$ and all other relative groups zero. The lower-middle Deligne formula gives
$$
\boxed{\dim IH_{\bar m}^i(\widehat M)=(1,0,1,1)\quad\text{for }i=0,1,2,3.}
$$
Thus degrees one and two have different dimensions, so there can be no three-dimensional <Poincare duality>. In the dual-lower-chain convention the dimensions are $(1,1,0,1)$, which likewise fail duality. Both computations exhibit the obstruction $H^1(T^2)=\mathbb Q^2\ne0$.
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