Separable Equation Validity When solving a "separable" ordinary differential equation, I've read that it's common practice to collect all terms involving the solution $y$ on one side, and all terms involving the independent parameter $t$ on the other side, and solve by integration.
$$\begin{align*}
\frac{dy}{dt}&=g(y)\,f(t)\\
\frac{dy}{g(y)}&=f(t)\,dt\\
\int \frac{dy}{g(y)}&=\int f(t)\,dt\,\,\,\,\,\,\textrm{(assume no stationary points with } y=0 \textrm{)}
\end{align*}$$
However, I can't see how this sort of thing is valid. What bounds do we use for integration? It seems to me that we are not invoking the same operator on both sides of the equation, given that the bounds of integration for the RHS can't possibly be the same as those for the LHS.
Also, how can one rigorously establish the validity of simply "multiplying by $dt$" on both sides?
 A: Well, the simplest way to verify this method is following. 
Rewrite the equation in this form:
$$ \frac{1}{g(y)} \frac{dy}{dt} = f(t). $$
If we plug a solution into this equation, we get
$$ \frac{1}{g(y(t))} y'(t) = f(t) $$
You see, we have a function of $t$ at the LHS and a function of $t$ at the RHS. Let's integrate them:
$$\int f(t)\, dt = \int \frac{1}{g(y(t))}\cdot y'(t) \, dt .$$
But what we have in second integral can be identified as a substitution $y \leftarrow y(t)$ that has been made in integral $\int \frac{dy}{g(y)}$.
And we finally get the formula
$$ \int \frac{dy}{g(y)}=\int f(t)\,dt .$$

What bounds do we use for integration?

It's possible to use both definite and indefinite integration here.
If you want to find a general solution of ODE $y' = g(y)f(t)$ 
then this formula 
$$ \int \frac{dy}{g(y)} =\int f(t)\,dt $$
gives it to you in form of implicit function. Since we have indefinite integration here, you might vary the value of integration constant and obtain different solutions to the equation. 
But when you want to solve the Cauchy problem $y(t_0) = y_0$ another formula would give you a solution:
$$ \int_{y_0}^{y(t)} \frac{du}{g(u)} =\int_{t_0}^{t} f(v)\,dv. $$

Also, how can one rigorously establish the validity of simply
  "multiplying by $dt$" on both sides?

This can be rigorously established by using the language of differential forms. See V.I. Arnold's Ordinary Differential Equations for example (page 20).
