I started seeing $$\int dx f(x)$$ in my freshman year of undergraduate.
It's pretty common and the more you learn about integration the more it makes sense.
Now, regarding this part:
$$\begin{align}\left(\int ^t_{t_0} dt' H(t')\right)^2\stackrel{?}{=}\int ^t_{t_0} H(t') dt'\int ^t_{t_0} H(t'')dt''\\\stackrel{?}{=}\int ^t_{t_0} dt' H(t')\int ^t_{t_0} dt'' H(t'')\\\stackrel{?}{=}\int ^t_{t_0} dt'\int ^t_{t_0} dt'' H(t') H(t'') \end{align}$$
All of the equals signs there are correct.
Integrals factor like this:
$$
\int dx \int dy \, f(x) \, g(y) =
\left( \int dx \, f(x) \right) \left( \int dy \,g(y) \right) \, ,
$$
which is all you did there.
In fact, these are all the same:
\begin{align}
\int \int dx \, dy \, f(x) g(y)
&= \int dx \int dy \, f(x) g(y) \\
&= \int dx \, dy \, f(x) g(y) \\
&= \left( \int dx \, f(x) \right) \left( \int dy \, g(y) \right) \\
&= \left( \int dx \, f(x) \right) \left( \int dx \, g(x) \right) \\
\end{align}
Note, however, that you cannot factor something like this:
$$
\int_0^t f(t') \left( \int_0^{t'} dt'' f(t'') \right) dt'
$$
because the limit of the second integral depends on the first integral's integration variable.
You can, however, write it as
$$
\int_0^t dt' f(t') \int_0^{t'} dt'' f(t'') \, .
$$
Operators
There is also an issue of when operators are involved:
$$\begin{align}\int dx \hat{F}(x) \hat{G}(x)\stackrel{?}{=}\int \hat{F}(x)dx\hat{G}(x)\\\stackrel{?}{=}\int \hat{F}(x)\hat{G}(x)dx\end{align}$$
There's really no difference.
The key is to remember that the $dx$ really doesn't mean anything other than to remind you which variable(s) in the integrand is being integrated.
By convention we tend to write the $dx$ either at the front or at the end.
I've never seen it written in the middle like that.
I think everyone would know what you mean, but putting the $dx$ is the middle of the integrands runs the risk that a reader won't notice them.
How do you know which operator is in the integrand?
Ok that's a good question!
It really comes down to the fact that notation has to be clear.
If you use the symbol $x$ to denote both an integration variable and a not-integrated variable, that's just asking for trouble.
It also shouldn't ever happen because integration variables are consumed by the integral, so they can't be referred to anywhere else in an equation.
For example, this makes no sense:
$$ g(x) = \int_0^1 \sin(x) dx$$
because there's no "free" $x$ on the right hand side.
And assuming the general case where $\hat{F}$ and $\hat{G}$ do not commute, you cannot write the integral with $\hat{G}(x)$ on the left of the integral. How is this not ambiguous?
Well, you certainly would not write
$$ \int dx \hat F(x) \hat G(x) \neq \left( \int dx \hat F(x) \right) \hat G(x) \, .$$
That just makes no sense.
A speech about functions, integrals, and notation
A function $f$ is a well-defined thing independent of any specific choice of variable.
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ takes one real number to another one.
It is, therefore, completely unambiguous to write an integral as
$$\int_0^1 f$$
with no $dx$.
If $f$ is the sine function, then we can write e.g. $\int_0^1 \sin$ with absolutely no ambiguity.
So why then do we so often write things like $\int_0^1 \sin(x) \, dx$?
Well, consider a slightly more complicated function like the function $f$ defined by the equation $f(x) = \sin(x)/x$.
How would we write this without variables?
Well, we'd name the inversion function $\text{inv}$ defined by $\text{inv}(x) = 1/x$, and then we could say $f = \sin \cdot \, \text{inv}$ and we would write the integral as $$\int_0^1 \sin \cdot \, \text{inv} \, .$$
That's a perfectly clear representation, but I think it's less common in practice for three reasons:
It's cumbersome to have to name every function. Imagine having to write $(\sin \circ \, \text{square}) \cdot \, \text{inv}$ instead of $\sin(x^2)/x$.
We often solve integrals by "variable transformation", and for some people it's easier to see what transformations to make if we represent the functions by their action on their variables.
If you want to evaluate a multi-dimensional integral, at some point you have to use Fubini's theorem and that's only really possibly once the integrals are expressed as nested one-dimensional integrals over separate variables.
Still, even with those three points, I do think that especially for gaining a better understanding of integration (e.g. the change of variables formula) it can be helpful to practice the "no $dx$ notation".
For example, I found that it was a key piece of my understanding how probability distributions transform under a change of variables.
The "no $dx$ notation" also makes a lot of sense for people with experience in programming because it has strong ties to the notion of a type system.
