= Solution
All <homology> and <cohomology> in this solution use $\mathbb F_2$ coefficients, so no <orientation> hypothesis is necessary. The <Poincare-Lefschetz duality> degrees are complementary:
$$
\boxed{H_k(W;\mathbb F_2)\cong H^{n-k}(W,\partial W;\mathbb F_2).}
$$
The starred notation in the question must be understood with this degree reversal.
Turn the <handle decomposition> upside down. An absolute $k$-handle becomes a relative $(n-k)$-handle based on $\partial W$. The absolute cellular boundary coefficient counts intersections of an <attaching sphere> with a <belt sphere>. In the reversed <handle decomposition>, the same intersections are counted in the opposite order. Over $\mathbb F_2$ the two incidence matrices are transposes, with no sign ambiguity. Thus the absolute <chain complex> is the complementary-degree dual of the relative <chain complex>. Taking <homology> and using the <universal coefficient theorem for cohomology> over a <field> proves the displayed <isomorphism>. This is <mod-two handle duality>.
To obtain a <perfect pairing> from the assumed closed case, form the <double of a manifold> $D(W)$ from two copies of $W$. The fold map $D(W)\to W$ is a <retraction>, so the inclusion of either copy induces an injection on <homology>. If $a\ne0$ in $H_k(W)$, its image in $H_k(D(W))$ is nonzero. Closed-manifold <mod-two Poincare duality> supplies a class $z\in H_{n-k}(D(W))$ with intersection $1$ against that image. Cut a representative of $z$ along the boundary and retain its part in the first copy; it defines $b\in H_{n-k}(W,\partial W)$. Equivalently, apply the quotient map $D(W)\to D(W)/W_-\cong W/\partial W$ and <excision>. Intersections with a representative of $a$ in the interior are unchanged, so $a\cdot b=1$. This proves nondegeneracy in the first variable. <Mod-two handle duality> gives equal finite dimensions for the two spaces, hence nondegeneracy in the second variable as well.
For the three-dimensional conclusion, set
$$
V=H_1(\partial W),\qquad L=\ker\bigl(H_1(\partial W)\to H_1(W)\bigr).
$$
The <long exact sequence in relative homology> says $L=\operatorname{im}\partial$, where $\partial:H_2(W,\partial W)\to V$. The closed-surface <intersection pairing> on $V$ is nondegenerate. A collar calculation gives the adjoint identity
$$
(\partial b)\cdot x=b\cdot i_*x.
$$
Indeed, push $x$ slightly into the collar: intersections with the relative surface representing $b$ correspond to its boundary intersections with $x$. By the absolute-relative <perfect pairing> already proved, $x$ is orthogonal to every $\partial b$ precisely when $i_*x=0$. Therefore $L^\perp=L$.
For any finite-dimensional space with a <perfect pairing>, $\dim L+\dim L^\perp=\dim V$. Hence the \b[half-lives-half-dies theorem] gives
$$
\boxed{\dim\ker i_*=\frac12\dim H_1(\partial W;\mathbb F_2).}
$$
But $H_1(\mathbb{RP}^2;\mathbb F_2)\cong\mathbb F_2$ has dimension one. It cannot be the entire boundary of a compact three-manifold, because the displayed dimension would be $1/2$. This excludes nonorientable three-manifolds as well as orientable ones.
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