Apparent paradox regarding spontaneity
The problem is that you are starting the derivation from conditions that ensure reversibility or spontaneity, but you are not considering conditions that lead to the third possibility.
The following expression from your post is generally true:
$dq-dq_{rev}=dw_{rev}-dw $
I've dropped the &quot;irreversible&quot; labels and assume that $dq$ and $dw$ refer to some process that you have dreamed up.
Now the condition that leads to non-spontaneous processes must be
$dw_{rev}-dw&gt;0$
since you have shown that the other conditions are for spontaneous processes, which leads to
$dq−dq_{rev}&gt;0$
and therefore
$dG&gt;0$
which is what you are looking for.
If you forget about $dG$ for a second and focus on  how you can get to the equation $dq−dq_{rev}&gt;0$ (since you already showed the next step to an inequality in terms of $dG$), you see that to arrive at this equation you have to say that the system can do more work in a process you dreamed up than in a reversible process  ($dw_{rev}-dw&gt;0$), which violates the 2nd law. To carry out a process in which $dG&gt;0$, you have to correct this violation and make up the work difference by having the surroundings do additional (more) work on the system than was assumed in the original analysis.
By the way, you should be careful using PV-work in your examples, as for an ideal gas in the absence of reactions, Boyle's law says that the pressure will change during an isothermal expansion and Charles' law says that the temperature changes in an isobaric process. In neither case would the Gibbs free energy provide the criterion for spontaneity.