How does differential equations in the space of operator valued functions work?

Question

In my Quantum Mechanics class, we talked about Schrödinger Equation for the time evolution operator $\hat U(t)$, $$i\hbar\frac{d\, \hat U(t)}{dt} = \hat H(t)\hat{U}(t),\qquad \hat U(0)=\mathrm{id},$$ with $\hat H(t)$ being some self adjoint operator on a Hilbert space $\mathcal H$, with the solution $\hat U(t)$ being some unitary operators.

And if $\hat H$ is independent of time, then we are given the solution $$\hat U(t) = e^{-i\hat H t/\hbar}.$$ We were told to just plug in and check that this is a solution, and were given no further explanation.

I'm just basically not sure how most of this works. More specifically, I have the following questions.

1. How is the derivative defined? My guess is that, we are using the operator norm for the operator, and define the derivative the normal way?
2. If there's an integral, how is it defined? I've only seen lebesgue integral for real and complex valued function, and I couldn't think of a way it could work for operators.
3. How do we know that the solution $\hat U (t)$ will remain unitary, provided the only thing we know is $\hat H(t)$ is self adjoint and $\hat U(0) = \mathrm{id}$.
4. If $\hat H(t_1)$ commutes with $\hat H(t_2)$, is it true that the solution is $\hat U(t) = e^{-i\int \hat H(t) \,dt /\hbar}$? My intuition says this should be true, as $\hat U(t)$ should always commute with $\hat H(t)$ in this case.
5. How much does the theory of normal differential equation carries over? Do we know that the solution exist? Is it unique? etc. The most general differential equation I've seen is for smooth functions on manifolds, and the one here is obviously very different (the functions aren't even commutative).
6. What are some good references for these (if there are any)?

Note: I'm an undergraduate student double majoring in math and physics, and I have taken several graduate courses. So don't expect me to know any of the advanced results, but I'd love to learn about them :)

Answer 1

I've found some relevant information in the text One-Parameter Semigroups for Linear Evolution Equations. I think it's only relevant for the case when $\hat H(t)$ is independent of time.

With some more reading, I believe the answer for question 1 and 2 is the Fréchet derivative and Bochner integral with the norm being the operator norm.

I wasn't able to find relevant information for time dependent $\hat H(t)$ though, and would love to see if anyone else can find anything.