**(1)** 
$H^2(K_3,\mathbb{Z})$ is a rank-22 unimodular lattice and can be decomposed as $2\,\overline{E_8}\oplus 3\,H_2$. 
Here $\overline{E_8}$ means the negative $E_8$ lattice.
$H_2$ means the rank-2 hyperbolic lattice.
The $E_8$ lattice is not diagonalizable but stably diagonalizable, i.e., $E_8\oplus \overline{I}\simeq \overline{I}\oplus 8\,I$, where $I$ denotes the positive rank-1 unimodular lattice.
Therefore, 

$$H^2(K_3,\mathbb{Z})\oplus2\,I\ \simeq\ \oplus 16\,\overline{I}\,\oplus 2\,I\,\oplus 3\,H_2\,.$$

According to Freedman's theorem, this means a homeomorphism between the two connected sums

$$K_3\ \#\ 2\,\mathbb{C}P^2\ \simeq\ \#\ 16\,\overline{\mathbb{C}P^2}\ \#\ 2\,\mathbb{C}P^2\ \#\ 3\,S^2\!\times\!S^2\,.$$

This probably fails to be a diffeomorphism.
But a stabilization theorem of C.T.C. Wall says that we can always obtain a diffeomorphism by adding sufficiently many copies of $S^2\!\times\!S^2$.
Namely, there is an integer $k$ such that we have a diffeomorphism

$$K_3\ \#\ 2\,\mathbb{C}P^2\ \#\ k\,S^2\!\times\!S^2 \ \simeq\ \#\ 16\,\overline{\mathbb{C}P^2}\ \#\ 2\,\mathbb{C}P^2\ \#\ (3\!+\!k)\,S^2\!\times\!S^2\,.$$

Now, considering the traces of the connected sums on both sides, we obtain an oriented bordism $W_1$ from $K_3\coprod 2\,\mathbb{C}P^2\coprod k\,S^2\!\times\!S^2$ to $\coprod 16\,\overline{\mathbb{C}P^2}\coprod 2\,\mathbb{C}P^2\coprod (3\!+\!k)\,S^2\!\times\!S^2$.
Viewed in an alternative way, $W_1$ is an oriented bordism from $K_3$ to

$$\coprod 18\,\overline{\mathbb{C}P^2}\coprod 2\,\mathbb{C}P^2\coprod (3\!+\!2k)\,S^2\!\times\!S^2\,.$$

It is then a trivial task to find an oriented bordism $W_2$ from the above to $\coprod 16\,\overline{\mathbb{C}P^2}$.
Hence the combination of $W_1$ and $W_2$ gives an oriented bordism you want.

You may find relevant materials in Alexandru Scorpan's book [The Wild World of 4-manifolds][1] and the references thereof.

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**(2)** I don't quite understand the question.
If you have a spin 5-manifold, it induces a spin structure on each connected component of its boundary.

[1]: https://people.math.ethz.ch/~dkosanovic/24-FS/Scorpan-ww4m.pdf