What is the rigorous justification for treating differentials as fractions while solving differential equations? For example, given the differential equation
$\dfrac{\mathrm{d}y}{\mathrm{d}x}=y\tag*{}$
We do,
$\dfrac{\mathrm{d}y}{y}=\mathrm{d}x\tag{*}$
At which point we go and integrate both sides.
Is there a rigorous justification for being able to go to $(*)?$
Basically you are using the fundamental theorem of calculus and the chain rule (combined together they constitute the rule of integration by substitution). When you write $f(y) dy = g(x) dx$ you are really writing $f(y(x)) \frac{dy}{dx} = g(x)$. You then integrate both sides over the same limits, obtaining 

$$\int_a^b f(y(x)) \frac{dy}{dx} dx = \int_a^b g(x) dx$$

The left side is

$$\int_{y(a)}^{y(b)} f(y) dy$$

so that "dropping the integral signs" gives

$$f(y) dy = g(x) dx.$$

But in the standard framework this last equation does not really mean anything.
The definition of "differential" is $dy = f'(x) dx$.  Replace $f'(x)$ with its other notation $\frac{dy}{dx}$ to get $dy = \frac{dy}{dx} dx$.  So when you start with 

$$\frac{dy}{dx} = y$$

you can multiply both sides by $dx$

$$\frac{dy}{dx} \; dx = y  \; dx$$

and then replace the left side by $dy$.