"What is a (spatial) frame of reference in quantum mechanics? When we make a transformation e.g. in a H-atom from the proton frame of reference to the electron frame of reference. The whole point of QM is that the electron cannot be treated as a dot somewhere in space but that it has an associated wavefunction and only acquires a certain position when I make an appropriate measurement. So how can I transform to the electron frame of reference making the origin to be the position of the electron if I don't know it?"

What is the coordinate in this system then and how do they all connect to each other? I've read the mentioning of an observer and the observer's state of motion and I don't understand how that relates to a frame of reference.

If you are traveling at $x$ speed the time will pass for you slower than to an observer that is relatively stopped. That's all just because a photon released at the $x$ speed can't travel faster than the $c$ limit.
What happens if you have two bodies, $A$ and $B$ moving towards each other. If $A$ releases a light beam, and $B$ measures it (the speed of the photons), the speed measured is still the same? The only difference will be the wave length? And if we have the opposite case, $A$ and $B$ are moving away from each other, we get the red shift, but the speed measured will be still the same?

Observer A and B are at the same "depth" in a gravity well. Observer B then descends into the well. A will observe B's time as going slower than their own. B will observe A's time as going faster than their own.
What happens if B were to ascend the well back to A's depth, would B's local time speed back up to the same rate as A's, but B would be younger (relative to A)?
What about the paradox caused by relative motion (ignoring gravity)? If A is moving relative to B, A and B will both observe the other's time as going slower. If A and B were together initially, then B moves away and returns, do their clocks agree?

Given a point mass, with $\underset{\_}{x}$ the position vector, on which acts a force $\underset{\_}{F}$ such that it is conservative:
$\underset{\_}{F}=-\mathrm{\nabla }U(\underset{\_}{x}).$
Then if I change frame of reference from a inertial one to a non-inertial one, it is true that the (total) force (meaning the vector sum of all the forces acting w.r.t. the new frame of reference) remains conservative?

Formula for the Bekenstein bound
$S\le \frac{2\pi kRE}{\hslash c}$
where $E$ is the total mass-energy. That seems to imply that the presence of a black hole in the region is dependent on an observer's frame of reference. Yet, my understanding is that the Bekenstein bound is the maximum entropy that any area can withstand before collapsing into a black hole.
Does this mean that the existence of black holes is observer dependent? Or that even if an observer does not report a black hole in their frame, one is guaranteed to form there in the future?

The relativistic energy-momentum equation is:
$E^2=(pc{)}^2+(m{c}^2{)}^2.$
Also, we have $pc=Ev/c$, so we get:
$E=m{c}^2/(1-v^2/c^2{)}^{1/2}.$
Now, accelerating a proton to near the speed of light:
$\begin{array}{}0.990000000000000& c=>& 0.0000000011& J=& 0.01& TeV\\ 0.999000000000000& c=>& 0.0000000034& J=& 0.02& TeV\\ 0.999900000000000& c=>& 0.0000000106& J=& 0.07& TeV\\ 0.999990000000000& c=>& 0.0000000336& J=& 0.21& TeV\\ 0.999999000000000& c=>& 0.0000001063& J=& 0.66& TeV\\ 0.999999900000000& c=>& 0.0000003361& J=& 2.10& TeV\\ 0.999999990000000& c=>& 0.0000010630& J=& 6.64& TeV\\ 0.999999999000000& c=>& 0.0000033614& J=& 20.98& TeV\\ 0.999999999900000& c=>& 0.0000106298& J=& 66.35& TeV\\ 0.999999999990000& c=>& 0.0000336143& J=& 209.83& TeV\\ 0.999999999999000& c=>& 0.0001062989& J=& 663.54& TeV\\ 0.999999999999900& c=>& 0.0003360908& J=& 2,097.94& TeV\\ 0.999999999999990& c=>& 0.0010634026& J=& 6,637.97& TeV\\ 0.999999999999999& c=>& 0.0033627744& J=& 20,991.10& TeV\end{array}$
If the LHC is accelerating protons to $7TeV$ it means they're traveling with a speed of $0.99999999c$.
Is everything above correct?

For a car that is accelerating linearly, the non-inertial frame of reference is the driver in the car where from his reference frame, the car is stationary. It is so called stationary because the non-inertial frame of reference has the same acceleration as the car. Is like the car's acceleration "transform" the driver frame of reference into a non-inertial. That's why in the non-inertial frame of reference, there is no force acting on the car.
But when the car is driving in circles at a constant speed, in the non-inertial frame of reference there is a force acting on the car, which is the centripetal force. Why isn't this frame of reference like the above, not having the acceleration found in their each respective inertial reference frame? Why can't we have a non-inertial reference frame(due to rotation) whereby there is no centripetal force, subsequently eradicating the need for a centrifugal force?