What precisely is the wave function a probability density of? In QM the norm of the wave function $\psi(\vec{x})$ is said to be the probability density that the particle is at $\psi(\vec{x})$ if one would observe its position. Generally, nothing more is explained what this exactly means, yet I'm confused by it. 
Hence my question: Does it mean that any given time, say, the electron in ground state of Hydrogen is at some exact position $\vec{x}$, but we just do not know exactly where, unless we measure it? 
If so, are such measurements actually done, e.g. was the location of the electron found at location $\vec{x}$, say, so many Angstrom from the nucleus at these polar angles in a particular measurement (given if repeated many times it should follow the distribution given by the wave function).
Note: The interpretation the electron is extended in space while not intuitive would not be that confusion to me but yet I always see it described as a probability density of being located at some $\vec{x}$ which implies it can be observed to be at some exact location.
 A: To be precise, it is not the complex-valued wave function $\psi(\vec{x})$ that is interpreted as a probability density, but its absolute square $|\psi(\vec{x})|^2$ is, sometimes $\psi(\vec{x})$ is called probability wave to stress the difference. This is significant, if $\psi(\vec{x})$ itself was a density, or if $|\psi(\vec{x})|^2$ itself obeyed an evolution equation, classical statistical interpretation of them as particle densities might have been possible, as in the Fokker–Planck equation. As it is, the probabilistic interpretation does not mean that electron is at some exact but unknown position at some time $t$, it only means that if a measurement is performed at that time, then the probability of detecting it in some region is equal to the integral of $|\psi(\vec{x})|^2$ over that region, see the Born rule. 
The "measurement" roughly means that the electron is made to interact with another system, which transforms its wave function into an approximation of one of the eigenfunctions of the position operator (which are $\delta$ functions). It makes no sense to ask where electron "is" while it is in a superposition state $\psi(\vec{x})$ (it is a superposition of it being everywhere), only for position eigenstates  the question is meaningful. So it makes no sense to ask where the electron is unless the measurement is actually performed (or the state happens to be localized for some other reason). Only half of the "measurement" process is non-controversially understood, see decoherence. The other half, namely how the eigenfunction in question is selected, is answered differently by different interpretations of quantum mechanics. In the Copenhagen interpretation, for example, the measurement indeterministically "collapses" the wave function into one of the eigenfunctions, and integral of $|\psi(\vec{x})|^2$ gives the probability of the collapsed eigenfunctions being localized in a particular region.
To determine the position of an electron it must be probed, the more precision is sought the stronger the probing interaction must be. One can bombard an atom with particles to knock an electron off of its orbit, then detect the scattered electron and infer its position by backtracking the evolution. Something like this is done in photoionization microscopy, which was used by  Stodolna et al. in 2013 to image the hydrogen atom's wave function (but they did not do it by knocking off one electron at a time). Or one can scatter the particles off of the atom and detect them, as Rutherford did. Neither is sensitive enough to pinpoint individual electron locations within an atom, usually only statistical data is collected. Even if they did the atom would be destroyed as a result of such "pinpointing" due to the required interaction energies, so the measured location will no longer be "within the atom" after the collapse. So-called nondemolition measurements, which preserve the atom, would not be able to do the pinpointing, see What is the experiment used to actually observe the position of the electron in the H atom?
A: The electron is an  elementary particle in the standard model of particle physics. As such it is postulated to be a point particle. BUT this is quantum mechanics, the term particle is just a tag on a quantum mechanical entity. 
Quantum mechanical entities follow boundary conditions in completely determined solutions of quantum mechanical equations. Within these solutions they are described by a function  F(x,y,z,t) but this is a complex function and can acquire a physical meaning within an expectation value of a position operator or a momentum operator, which involves the complex conjugate squared,. It is a postulate of quantum mechanics that this is a probability density for finding the electron. One has to do many measurements with the same boundary conditions to get the probability distribution.
In the case of the double slit single electron at a time one can see the probability density developing because the electrons are not in a bound state. A single electron interacts with the screen and leaves a point characteristic of a classical particle. It is not spread out over the whole pattern. 
Electrons in atoms are described by the orbitals and as Conifold has referred also the Hydrogen orbitals have been measured in a specific experiment:

It is a probability density distribution, a collective measurement of interactions with photons

After zapping the atom with laser pulses, ionized electrons escaped and followed a particular trajectory to a 2D detector (a dual microchannel plate [MCP] detector placed perpendicular to the field itself). There are many trajectories that can be taken by the electrons to reach the same point on the detector, thus providing the researchers with a set of interference patterns — patterns that reflected the nodal structure of the wave function.

The x and y coordinates are in mm on the detector plane.
The color coding is the intensity registered at the detector. A point on the plot builds the interference patterns of the  position of the electrons. In a sense , the double slit interference pattern reflects the geometry of the two slits and one could arrive at the geometry by using the pattern. The interference pattern seen above can be  correlated with the calculations of the orbitals.  For details the paper can be seen here.
From this data an estimate within the Heisenberg uncertainty could be made of the position of the probable d(V) of the electron in the atomic dimensions but as the hydrogen atom has a complete quantum mechanical solution, one uses the mathematics of   orbitals.
