Yes, here is a little more detail:
Within the nucleus, there exists a shell structure to the protons and neutrons inhabiting it, similar to the shell structure of the electron orbitals outside the nucleus. Those different shells contain nucleons with different energies. At the same time, there is a natural tendency for those protons and neutrons to in some sense "agglomerate" into alpha particle associations, because of the extremely high binding energy of such clumps.
But it isn't geometrically possible for protons and neutrons to 1) pack themselves down perfectly into a hexagonal-close-packed ("fully dense") form, 2) maintain the alpha particle associations, and 3) obey the shell structure rules as you add more and more nucleons to the nucleus. Compromises are necessary; this means that there will be certain nuclei in the periodic table that are lucky and have high binding energies and others that are less tightly-bound, and still others which are so unstable that they decay quickly and are not found in nature.
In a practical sense, this means you can model alpha decay as a highly asymmetrical case of fission, which one of the fission products is an intact alpha particle i.e., a helium nucleus.
This also means that there are nucleon configurations (or packing schemes) which are energetically unfavored but which can be created through collisions with other particles, yielding something called a shape isomer which you can think of as a metastable nucleus with a lump sticking out of one side or having a football-like shape instead of being spherical. The shape isomer can have a half-life for decay of microseconds, seconds, minutes, hours, days or years, and can produce (for example) a highly-energetic gamma ray when it decays.
A certain isotope of technetium, for example, can be produced in an accelerator to yield a shape isomer with a decay half-live of order ~6 hours and which produces highly penetrating 140 kiloelectron-volt gamma rays. These are routinely prepared and used as tracer elements in various medical procedures.
It's definitely not the case that the electron cloud is undisturbed. The process is quite violent on the scale of the atom undergoing alpha emission.
The electron cloud is excited by the process of the emission of the alpha particle, so the remaining ion isn't in its ground state. Moreover, the remaining nucleus also gets some recoil from the alpha particle, which also adds to disturbance of the electron cloud. The disturbance may in fact lead to some electrons being ejected from the ion.
For example, here is a 1975 paper which reports the "shake-off" of electrons from polonium-210 nuclei during the alpha decay to lead-206. The experiment observed simultaneous alpha particles, emitted electrons, and x-rays from the filling of the "hole" in the lead's electron cloud. Current best results suggest about ten or fifteen electrons are ejected per million polonium alpha decays.
Why it's usually "conveniently ignored" is because chemistry is rarely of interest when discussing nuclear physics.
Best Answer
Definitively 4 distinct nucleons. Combinations of more than 4 quarks have never been observed. The existence of tetraquarks is pretty much confirmed [1]: the so-called Z(4430) whose quark content is $c\bar{c}d\bar{u}$. The next-lightest candidate, the pentaquark, has been entertained but the conclusion is currently that it does not exist. So 12 quarks!
Interestingly, note that the tetraquark mentioned above is heavier than an $\alpha$ particule (4.4 vs 3.7 GeV/c$^2$). The putative pentaquark resonances have masses around 4.4 GeV/c$^2$ too. Thus even without all the evidences provided by a century of nuclear physics which point to the fact that $\alpha$ particle are made of nucleons, clearly a dodecaquark would be far too heavy…
[1] https://arxiv.org/abs/1404.1903