1) You can of course write down these amplitudes in any basis you choose, as long as you take into account matrix elements of the CKM and PMNS matrix.
2) There is indeed a difference here, neutrinos are produced exclusively by the weak interaction, whereas quarks can be pair produced by the strong (or electric) force, or produced by weak decays.
3) in principle hadronic states can oscillate, $K-\bar{K}$ and $B-\bar{B}$ oscillations have been studied in great detail. Also, interference effects in weak decays certainly exist. There are some difficulties, however, with observing''$d-s$'' oscillations: i) Quarks are confined, so the actual oscillation would have to be $\pi-K$. ii) But pions and kaons are (typically) produced as strong (mass) eigenstates. iii) The mass difference is quite large, and the oscillation length would be very short. iv) Kaons are unstable (not necessarily a problem, they live sufficiently long to observe kaon-anti-kaon oscillations).
I gather you are asking about similarities and differences between hadronic and neutrino neutral particle oscillation. There are so many questions packed within your units, that I'll try to answer isolated facets of the jumble, apologizing, with Pascal (Je n’ai fait celle-ci plus longue que parce que je n’ai pas eu le loisir de la faire plus courte), that I lacked the time to make the answers shorter; and leave it to you to fit the facets together.
For starters, both oscillation phenomena are due to quantum-mechanical interference of complex propagation amplitudes, and are only possible due to QM. In both cases, the propagating Hamiltonian (mass) eigenstates ($K_L,~ K_S$, $B_L, ~B_S$; $\nu_{1,2,3}$) are not the interaction combinations which production mechanisms and detectors rely on, so the mismatches in time and hence distance of the ever-shifting interaction combination states are described as an oscillation.
(There are theological discussions about preferences in momentum vs energy eigenstates involved in the parallel propagation, but just don't go there.)
The differences between the two cases are just in the microscopic QM description of these amps. You wrote down the diagrams for Bottomness oscillations of mesons.
Why are these processes called flavor-changing neutral currents? It is because there is no electric charge transfer? So what are examples of flavour changing charged currents?
The standard plinth of the QFT edifice is the current, in this context a bilinear of quarks coupling to mesons, for example, and to something bosonic, like a Z or a W, or a grouping $W^+W^-$. The net charge of the object makes the difference between a charged current, versus a neutral current. In your case, the neutral currents involved are
$\bar d \Gamma b_L$ and $\bar b \Gamma d_L$ , where I've left the tensor structure of Γ vague/open, and you may construct them by pinching charged currents, as in your loop diagrams. They are not flavor neutral, so the amp you wrote changes bottomness by two. Because of the CKM/PMNS construction, charged currents can and do change flavor, on account of the built-in intergeneration straddling of the W couplings.
Are processes involving the $Z$ boson allowed? (I expect the answer is no because the weak neutral interaction couples fermions of the same type)
Right you are. If the neutral current couplings to the Z were not diagonal in the mass eigenstates, the unitary basis change to those would cancel itself in the couplings. But now you have exhausted this freedom, and there is nothing left to make the above bottomness-violating term flavor diagonal.
In the lepton sector, there is the $U_{PMNS}$ matrix. So, is it also possible to have diagrams where "oscillation" with flavour violation can occur (e.g. we have internal lepton line and external lines with neutrinos)?
Yes, this is the point behind neutrino oscillations, except you got the internal particle wrong (the "oscillating"/propagating particle has to be neutral, so the neutrinos, as above), and you don't need a loop: it is a tree diagram. Here is a trail map on how to convert the snipped upper part of you second diagram to a neutrino production+oscillation+detection one.
A $\pi^+$ decays at the production vertex to a virtual $W^+$ which converts to $\nu_\mu$ and $\mu^+$, lost at production. But $\nu_\mu$ is a fiction: it is a superposition of the propagating $\nu_1,\nu_2,\nu_3$, zipping along with infinitesimally small relative delays. At the detection vertex, they suck up a virtual $W^-$ off the detector's nuclei, to turn into a negative lepton, but, this time, on account of the mismatch of velocities in the wave packet, a $e^-$, for the sake of argument. So, this presents as an oscillation ( ⇄ leptonic flavor change) of the bogus $\nu_\mu \to \nu_e$. You are not studying mesons, so you don't need two fermion lines as in B oscillations, one fermion line propagation will do.
Why is neutrino oscillation instead described quantum mechanically?
Explained above: it is all QM.
Is it because we need not suppress the amplitude (unlike the GIM mechanism)?
Actually yes, in a way, if by this you mean "what makes the respective oscillations rare/slow enough to be detectable at macroscopic scales?" Indeed, (do the estimate calculation!), in the hadronic case, the doubly week, GIM suppressed, transition amp is sufficiently slow to give you a macroscopic (lab) effect for the high masses involved; while, for neutrinos, the tree amp suffices, on account of the smallness of neutrino masses: The relevant parameter for the distance of oscillation is L ~ E/m², long baseline.
Best Answer
My understanding of this question is really two different questions. Let me answer each of these in turn.
1) What is the relation between the CKM and PMNS matrices?
To see how this works consider the relevant quark interaction terms without any choice of basis, \begin{equation} - m _d \bar{d} d - m _u \bar{u} u - i W _\mu \bar{d} \gamma ^\mu P _L \bar{u} \end{equation} Here $ m _d $ and $ m _u $ are completely arbitrary $ 3 \times 3 $ matrices.
We can redefine the down type quarks such that $ m _d $ is diagonal, $ d \rightarrow U _d d $. This matrix can then be reabsorbed into $ u $ (by a choice of basis for $u$) keeping the charged current diagonal. However, after this second redefinition we can't redefine the up type quarks again since we lost that freedom.
Therefore to have mass eigenstates we must introduce a mixing matrix which we call the CKM (this is often referred to as a product of the transformations on the down-type and up-type quarks but this is a bit unnecessary since we can always redefine one of either the down-type or up-type quarks to be in the diagonal basis). The CKM appears in the charged current interaction, \begin{equation} W _\mu \bar{d} \gamma ^\mu P _L \bar{u} = W _\mu \bar{d}' \gamma ^\mu P _L V _{ CKM}\bar{u} ' \end{equation} Then we define a quark to be the mass eigenstates. The "cost" of this is that then we have to deal with uncertainty about which particle is produced in the charged current interaction since now particles of different generations can interact with the charged current. Its important here to note that this would not have been true if we called our ``quarks'' the fields that had a diagonal charged current.
That being said lets contrast with the charge lepton sector. Here we have, \begin{equation} - m _\ell \bar{\ell } \ell - m _\nu \bar{\nu } \nu - i W _\mu \bar{\ell } \gamma ^\mu P _L \bar{\nu } \end{equation} If the neutrinos were massless ($ m _\nu = 0 $) then we can just redefine the charged lepton basis such that their mass matrix is diagonal and we don't introduce any mixings into the charged current. However, if neutrinos do get a small mass then we have a choice we can diagonalize the neutrino matrix or leave the charged current diagonal.
On the other hand, unlike for the quarks, the mass eigenstates of the neutrino are almost impossible to produce. We have very little control over the neutrinos and they are typically made in one of the interaction eigenstates (in the basis in which $ m _\nu $ is nondiagonal), due to some charged current interaction. Thus the neutrinos are going to oscillate between the different mass eigenstates due to the state being in a superposition of energy eigenstates. Since we can't produce these mass eigenstates it is more convenient to call our "neutrinos" the states which we produce and let them oscillate.
Finally note that we often do diagonalize the neutrino matrix and define the analogue to the CKM known as the PMNS matrix, however this is more of a convenient way to parametrize the neutrino mass matrix then anything else.
2) Do quarks experience particle oscillations?
In general whenever the interaction eigenstates are not equal to the mass eigenstates particles can experience oscillations. In practice whether or not these oscillations are observable will depend on the interactions of the outgoing particles. Quarks interact significantly with their environment making their oscillations not observable in a physical experiment. To see how this plays out consider some collider producing down-type quarks (this can be say from top decays). The outgoing states will take the form,
$$|\rm outgoing\rangle = \#_1 |d\rangle +\#_2 |s \rangle+\#_3 |b \rangle$$
with the different coefficients determined by the CKM angle. When acted on by the time evolution operator, this state will mix into the other interaction eigenstates and hence when $|\rm outgoing \rangle$ propagates, it oscillates.
However, once these states are produced they are quickly "measured" by the environment through the subsequent processes such as showering and hadronization. The timescale for hadronization is $\Lambda_{QCD}^{-1} $ or a length scale of about a femtometer. This is way shorter than where we could place our detectors to see such oscillations. Once hadronization takes place the states decohere and quantum effects are no longer observable. Hence the linear combination is destroyed well before these particles are allowed to reach our detectors.