= Solution
\b[True: every integer occurs.] The simple-connectivity assumptions in parts 1 and 2 are local to those statements and are not imposed here. For a connected closed oriented four-manifold, <Poincare duality> gives
$$
\chi(M)=1-b_1+b_2-b_3+1=2-2b_1+b_2,
$$
since $b_3=b_1$. This expression has no fixed sign or parity.
Use $\chi(\mathbb{CP}^2)=3$, from its three even-dimensional cells, and $\chi(S^1\times S^3)=0$, by <Euler characteristic of a product>. The <Euler characteristic of a connected sum> in dimension four is
$$
\chi(A\#B)=\chi(A)+\chi(B)-2.
$$
It follows either by <Mayer–Vietoris sequence> after deleting balls, or by the intermediate-degree cohomology splitting and the single top and bottom generators. Therefore a <connected sum> of $r$ projective planes and $s$ copies of $S^1\times S^3$ has
$$
\chi=2+r-2s.
$$
Given an integer $h$, choose an integer $s\geq\max\{0,\lceil(2-h)/2\rceil\}$ and put $r=h-2+2s\geq0$. This produces a connected closed oriented four-manifold of <Euler characteristic> $h$. When $r=s=0$, use $S^4$. Thus
$$
\boxed{\{\chi(M):M\text{ connected, closed, oriented, }\dim M=4\}=\mathbb Z.}
$$
If simple connectivity were added, the possible values would instead satisfy $\chi=2+b_2\geq2$. This distinction is recorded in <Euler characteristics of closed oriented four-manifolds>.
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