According to the second law of thermodynamics, the total entropy of the Universe must always increase after any interaction (as I understand). So in the hydrogen atom, the electron has a high probability of being found in some region around the nucleus specified by its quantum number. My question is, if the hydrogen electron is probed, does this action increase the entropy of the atom's surroundings? If so, does this mean that the process of narrowing down the position of the electron (within the possible limits) increases the region over which one or more of the particles in the environment is likely to be found (i.e. does the increase in entropy imply that the 'probability cloud' of a particle expands over a larger region?)
To begin with, entropy is a classical thermodynamics concept. Different statistical frameworks , assuming some postulates, can define an entropy.
The basic formulation of entropy defined by statistical mechanics
where kB is the Boltzmann constant, equal to 1.38065×10^−23 J K−1. The summation is over all the possible microstates of the system, and p_i is the probability that the system is in the ith microstate.
where rho is the density matrix and ln is the matrix logarithm. This density matrix formulation is not needed in cases of thermal equilibrium so long as the basis states are chosen to be energy eigenstates. For most practical purposes, this can be taken as the fundamental definition of entropy since all other formulas for S can be mathematically derived from it, but not vice versa.
if the hydrogen electron is probed, does this action increase the entropy of the atom's surroundings?
The hydrogen electron can be probed with a photon of an appropriate energy for a transition to a higher level, or for ionizing the atom. The entropy of the whole sample is increased by these interactions, as different microstates are created which will add to the count. The probability cloud, the orbital, will change, the whole atom will have a different momentum etc. In the image, a higher energy level will have a larger space occupation for the probability.
On the other hand when the electron falls back to its ground state emitting one or more photons, again the entropy will increase because the number of microstates will go up, even though the lower orbital will occupy a smaller volume in space. Thus the increase in entropy does not depend on the spatial distribution of the electron orbitals, but on the number of microstates , so there is no one to one correspondence of entropy and the spatial probability "cloud"/orbital.
Yes. Any process that creates or changes information involves a local change in entropy, and requires energy to accomplish.
In quantum statistical mechanics entropy is not defined via the probability density of a single state but through the density matrix which talks about the "non-quantum uncertainty" over the states. This is the "probability density" over which all the entropy theorems are proven in the quantum world.
That is, if we know the system to be in a sharp quantum state, it's thermodynamical entropy is undefined (as is fittingly stated in the comment by Ján Lalinský), and it's quantum information entropy (defined by a certain trace of the density matrix) is zero. And this is true even after the action of measurement collapses the state to a different one. Entropy in the sense of thermodynamics is thus eliminated from this discussion.
But, to answer your question in a different way, we can talk about the growth of true uncertainty in observables. In this sense, by the collapse you either increase or decrease uncertainty with the lower bound set by the Heisenberg uncertainty principle. However, it is pretty clear from the linearity of quantum mechanics that e.g. for a superposition of two energy levels, the uncertainty will remain the same after collapsing a large ensemble. (Yeah, there's the bit on probability being quadratic, but this is no problem as long as we use quadratic uncertainties.)
To conclude, as long as we consider large ensembles, quantum mechanics doesn't really need a treatment of the measurement as any kind of physical action rather than as a curious information process. So the answer is no, the surroundings of a hydrogen atom need not in principle increase it's entropy in a measurement.