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.
The CKM, PMNS matrices are mathematically absolutely analogous, except that the values and even hierarchies of all the parameters are entirely different in the two cases. (Also, we don't know whether right-handed neutrinos exist and whether the effective Majorana masses may be derived from Dirac masses or something else.) But both matrices may be reduced to three "real angles" and one complex "CP-violating phase".
And the decay $W^+\to e^+\nu_\mu$ is entirely possible if $\nu_\mu$ actually represents a mass eigenstate of the neutrino, much like the convention we use in the case of quarks.
However, the difference is that the "oscillations" between quarks that are not mass eigenstates are extremely fast, and happen at the length scale of the nuclear radius, while the oscillations between flavors of neutrinos take hundreds or kilometers or much more. The difference arises because the neutrino mass matrix entries are hugely lower than the quark masses – even the general elements of the quark mass matrix. The strange quark's mass is 150 MeV while the neutrino masses are comparable to 1 meV. That's a difference of some 11 orders of magnitude. The "in practice" difference is actually much higher because the neutrinos always move nearly by the speed of light and their oscillations are therefore slowed down by the relativistic time dilation (much greater than for the quarks).
For this reason, because of the very slow neutrino oscillations, we don't actually use the neutrino mass eigenstates in discussions about particle physics processes. By $\nu_\mu$, we really mean the $SU(2)$ partner of the left-handed muon (the middle mass eigenstate in the 3D space of the charged leptons of a given charge), and then the decay to $e^+\nu_\mu$ is strictly prohibited by the $SU(2)$ gauge symmetry.
But the process $W^+\to e^+\nu_2$ where $\nu_2$ is a mass eigenstate of the neutrinos is allowed because $\nu_2$ contains a nonzero addition of $\nu_e$, and it's this process involving $\nu_2$ that is analogous to the flavor-changing processes with quarks (controlled by the Cabibbo angle etc.).
Best Answer
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).