(3) and (4) are false in general, even if we weaken $>$ to $\geq$. Let $\zeta:=e^{i\pi/8}$ be a primitive $16$-th root of unity, and let $\chi$ be the unique primitive Dirichlet character modulo $17$ satisfying $\chi(3)=\zeta^5$. Then $\chi(-1)=-1$, and
$$(\chi(1),\chi(2),\chi(3),\chi(4),\chi(5),\chi(6),\chi(7),\chi(8))=(1,\zeta^6,\zeta^5,\zeta^{12},\zeta^9,\zeta^{11},\zeta^7,\zeta^2).$$
However,
\begin{align*}Q(\chi)&=\sum_{k=1}^{16} k\chi(k)=\sum_{k=1}^{8} (2k-17)\chi(k)\\[6pt]
&=-(15+13\zeta^6+11\zeta^5+9\zeta^{12}+7\zeta^9+5\zeta^{11}+3\zeta^7+\zeta^2)\\[10pt]
&\approx \ 8.84701161719 - 4.91203840222\,i.
\end{align*}
So $Q(\chi)$ has positive real part, even though $\chi(-1)=-1$. This contradicts (3). Similarly, if we change $\zeta^5$ to $\zeta^3$ in the definition of $\chi$, we get a counterexample to (4).