So I saw that $d(xy) = 0 \to xy = c$.
This seems very intuitive, as if we were to put an integral sign before both expressions in the first equation, we get the result quite quickly.
My question is why are we allowed to stick integral signs before a differential? From my understanding, you can must integrate both sides with respect to some differential (e.g. integrate with respect to x), so is it not incorrect logic to just stick integral signs before terms?
On
Given the plot of your story, the RHS $xy=c$ means nothing. When $x$ and $y$ are real values, then $c=xy$ is another real number; so what.
It could be that the "variables" $x$ and $y$ depend on some hidden variable $t$ ("time"). In such a case we are talking about the function $$f(t):=x(t)\,y(t)\ .$$ The statement $df=0$ then is equivalent to $f'(t)={d\over dt}\bigl(x(t)y(t)\bigr)\equiv0$, and this means that there is a $c\in{\mathbb R}$ with $x(t)y(t)\equiv c$.
But it could also be that the function $$g(x,y):=xy$$ of the two variables $x$, $y$ is at stake. Then $dg=0$ defines at each point $(x,y)\ne(0,0)$ a tangent direction, and that's it.
Without more instructions,
$$d(xy)=0\cdot d(xy)$$ holds for all values of $xy$ and you can integrate both sides on $xy$, giving
$$xy+c=c'$$ or simply $$xy=c.$$
You can apply the same operations to both members of an identity, it remains an identity.