What determines the Hamiltonian of an isolated quantum system? Let an isolated quantum system be in state $|\psi\rangle$. Then, quantum mechanics says the system evolves in time according to some Hamiltonian $H$, which does not depend on $|\psi\rangle$. But the first postulate of quantum mechanics also says that $|\psi\rangle$ completely describes the isolated system. If so, how can $H$ not depend on it? What then determines $H$ of an isolated system? (Take the universe as an example, that's isolated in the ideal sense, as there is nothing outside it. Still, its Hamiltonian depends on something other than its own state?)
 A: 
But the first postulate of quantum mechanics also says that |ψ⟩ completely describes the isolated system.

$|\psi\rangle$ completely describes the state of an isolated system at that moment in time.  It does not describe how that state evolves with time.

What then determines H of an isolated system?

One could ask the same about a classical system as well.  Different Hamiltonians describe different physical systems.  If you have a particular system in mind, which Hamiltonian you should use to model it is not always a trivial question to answer.
When describing particles interacting with some potential, you will typically encounter so-called Schrodinger Hamiltonians of the form
$$\hat H = \frac{1}{2m} \hat P^2 + V(\hat X)$$
with $\hat P$ and $\hat X$ the position and momentum operators defined on whichever Hilbert space is appropriate (usually $L^2(\mathbb R)$ for a 1D system or $L^2(\mathbb R^3)$ for a 3D system).
On the other hand, sometimes your system is best modeled as a system of interacting spins with fixed spatial locations as in the Ising model, which involves a different Hilbert space and a different kind of Hamiltonian.  Sometimes your system can be reasonably well-modeled as a simple two-state system subjected to an external potential, as in the ammonia maser toy model.
At the end of the day, constructing a mathematical model for a quantum system involves choosing a Hilbert space (which defines the set of states the system can occupy) and a Hamiltonian operator (which defines how those states evolve with time).  In real applications, both tasks are generally non-trivial; the standard recipe is to make those choices based on physical intuition and experience, check the predictions of the resulting model against experiment, and then update your choices if those predictions aren't sufficiently accurate for your needs.
A: The thing that the Hilbert-space vector $\vert \psi\rangle$ describes is, as the name suggests, the state of the system at a particular point in time. More precisely, $\vert \psi\rangle$ tells you about the statistics of any observable you'd want to measure at that point in time.
This has nothing to do with describing how the state of the system (i.e. $\vert \psi\rangle$) changes in time. That is described by the Hamiltonian, which is completely independent of the state of the system at some point in time.
To give you an example, let's say I have a spin $1/2$ particle in the $\left\vert\uparrow\right\rangle$ state. As an experimenter, I can obviously start tweaking my control instruments in order to manipulate the state of the system independently of the system's initial state. In fact, I might not even know what the state of the system  is to begin with! In this case, I'm changing the Hamiltonian of the system completely independently of what its initial state is.
