A big-picture conceptual answer, which I think is helpful before digging too deeply into the math:
Any quantum that is isolated (in a "box") from the outside world will remain in a superposition state and will never ``collapse." Even if there is a measurement device inside the box exists or the system "decoheres," to an outside observer, outside the "box" will never invoke the Born rule. What happens inside the box must be just deterministic propagation of the wavefunction (until it is open).
Put a quantum system and heat it to a billion degrees. Add as much decoherence as you'd like. Add little measurement devices in it. No matter what. If it is isolated from you (in a perfect "box"), it will not collapse. The Born rule will not occur.
So how does that resolve with the fact that an observer inside the box does see collapse? This is the "Wigner's Friend problem." The "solution" is that a person outside the box see's the observer inside the box as being in a quantum superposition of having measured both outcomes. Measurement only collapses the wavefunction in your perspective. It does not collapse in others perspectives if they do not make the measurement.
Decoherence is simply the continuous version of such a situation. Sometimes a quantum system can be "measured" by many other things (which we call "interaction with the enviornment"), to someone "outside the box", you expect to see a superposition of all the possible outcomes entangled with each individual outcome that was measured. And by opening the box you collapse the state and you check which outcome has occurred.
If you check a quantum system that hasn't interacted with anything, you can sometimes see weird interference fringes as the probability amplitudes interfere. As you allow for many different things to make measurements of this origionally isolated quantum system (all inside a box that you don't look at yet), then you will see that as you ramp up the interaction, when you open the box you will see less and less interference fringes.
There was a point where this decoherence process was thought by some to "solve the measurement problem" as it shows that interaction with the enviornment makes the interference fringes appear to disappear. But it is well understood now that essentially what's happening is that you end up with a messy mush of entangled states (that are in an even more complicted superposition state) and this mess is why is seems like the "quantum weirdness" through interference has disappeared.
User @acuriousmind goes through some of the technical outline of how one might show how this entanglement forms when a quantum system interacts with the enviornment through "pointer states," which are essentially little things that are making their own measurements of the state of the origionally isolated system through interaction.
My problem is that I have no idea how an external observer should look
like as a quantum system, and as a consequence I don't know how the
"meta-Hamiltonian" H would look like.
Because of these reasons there's a big difference between an external observer (outside the box) and an internal observer (inside the box).
An external observer, when making measurements will collapse the state, so you need more than the schrodinger equation. You would have to pair it with the Born rule and do some stocastic simulations. (This isn't so common)
Decoherence considers internal observers that are inside of the box, that are interacting with the origionally isolated system. This is simply unitary evolution of a larger Hamiltonian and an "observation" is literally just when the "measuring object" would have some information about the state on its own. Entanglement like $|0\rangle |H\rangle + |1\rangle |T\rangle$ shows that in the perspective of the second object, being in state H means it knows that the first object is 0. Since collapse never happens you simply just propagate the unitary evolution and you get your internal "observers."
- How does an internal observer (i.e., that is part of the universe)
"feel" the state of (the rest of) the universe?
An internal observer "feels" it by becoming a superpostion of measuring different outcomes.
An external observer "feels" it by experiencing itself in one of the possibilities of those outcomes. (This is why the many-worlds intepretation came out of a resolution to the Wigner's Friend problem. )
- Would two different observers always agree on the wave function that
models (the rest of) the universe? If not, what are the conditions to
ensure they do?
But maybe a simpler example to get what I think you are looking for:
Imagine my friend performs a measurement, and then tells me the result. Then this means that we are both correlated and you have what you want: two observers that always agree on the outcome that has occurred.
Now you might ask, how do I know that what I consciously experience is the same outcome as the outcome that you consciuosly experience, as it's possible that we experience wavefunction collapse different.
And the answer, I believe, is that particularly question is unanswerable unless you test it yourself.
- What is the true state of the universe? If the universe has its own
wave function, but there is no external observer that makes it
collapse, does it really make sense to use the wave function to model
the state of the universe? Or can something more precise be used, that
takes into account the experiences of the internal observers? (Edit:
here by “universe” I am not necessarily talking about the real
universe, I mean “a closed quantum system that contains observers”;
the question is about a quantum system as a model (with internal
observers) and it is vague because I cannot make it more rigorous).
"but there is no external observer that makes it collapse, does it really make sense to use the wave function to model the state of the universe?"
To an external observer outside the universe that has not opened the box of the universe since the beginning of time: the universe is in a superposition of all possible outcomes and wavefunction collapse has never ever happened. It is in a superposition of all possible outcomes that could ever probabilisitically happen, and if you are to interpret these different outcomes as real then you believe in the many-worlds-interpretation of QM.
Or can something more precise be used, that takes into account the
experiences of the internal observers?
This is maybe a bit more technical than what you are looking for but:
Something precise about the experience of others cannot be used.
I think you are flip-floping external and internal, but if you want to know what can happen in the perspective or experience of an observer, you cannot know what others will experience. In fact, I have written about how there can be other rules for what other observers experience and you would not be able to measure them (see here).
But you can say something loose like "it seems like other observers, in my perspective, will tell me that they observe the Born rule," which may or may not be satisfying to you.