This is not a common question at all. But, it is a rather obvious one and props to you for taking the time to think about it because asking it is a good frame for exploring parts of the Standard Model and particle physics that most people don't spend much time thinking about in an enjoyable way.
It is non-trivial to figure out just what such a world would look like, but we have almost all of the information we need to figure it out from what we know about the Standard Model so long as the assumptions are clarified. Certainly, hadrons with third generation quarks would be as scarce as they are in our world in a world with second but not first generation quarks. And, only a very small number of hadrons containing only charm and/or strange quarks would be stable. The trick would be to first figure out which hadrons were stable or metastable. This, in turn, would depend upon whether charm quarks could decay to strange quarks in the same manner that down quarks can decay to up quarks in our world (with the reverse also occurring in each respective scenario when not barred by conservation of energy), or whether charm and strange quarks simply could not decay further even though that is not strictly analogous to the first generation particle behavior. I'll focus on the former, because it is closer to the concept that I think you're trying to explore and involves dynamics that are pretty easy to make sense of. Even a slight deviation from the Standard Model's weak force makes the question a lot more foreign and complicated when simply eliminating the option of first generation fundamental fermions without specifically specifying that they are stable creates the kind of stability that I think you are envisioning.
Electromagnetically, a spin-1/2 baryon with two charm quarks and a strange quark, or with one charm quark and two strange quarks, have the same basic chemical properties as protons and neutrons respectively. But, the proton and neutron have almost identical masses, while the second generation equivalent would not, so baryons with charm quarks would be strongly prone to decay, if possible.
The charmed omega baryon which is analogous to the neutron has a mass of about 2.7 GeV, and the doubly charmed omega baryon which is analogous to the proton has an unknown mass which would be at least about 3.9 GeV. Several spin-3/2 baryons would be lighter than the spin-1/2 baryons made up only of charm and strange quarks. The omega baryon is spin-3/2 and has a mass of about 1.7 GeV and an electromagnetic charge of -1. A spin-3/2 charmed omega baryon (analogous to the positively charge delta baryon in our world) has a mass of about 2.8 GeV. The mass of a spin-3/2 double charmed omega baryon (analogous to the neutral delta baryon in our world) has an unknown mass but would be at least 4.0 GeV. The mass of a triply charmed spin-3/2 omega baryon (analogous to the charge +2 delta baryon in our world) has an unknown mass but would be at least 5.2 GeV.
If charmed baryons were allowed to decay to less charmed baryons in this world, almost all matter would decay to uncharmed omega baryons, atypical isotypes of atomic nuclei would be very rare, and every atom would have a negative electric charge. Assuming that the W+ bosons produced in charm quark ultimately decayed to produce mostly antimuons and muon neutrino pairs (plausible given conservation of charge), the resulting atoms would have chemistry a lot like atoms in our own world but with only one isotype of each periodic table element and atomic masses a bit more than twice as great.
The fact that baryons in atoms would usually have spin-3/2 rather than spin-1/2 would also tweak the chemistry, in ways that would be predictable but subtle. No spin-3/2 particles in our world have mean lifetimes of more than a fraction of a second, so we don't have a lot of experimental data on how chemistry would differ in a world of spin-3/2 rather than spin-1/2 particles. But, since both are fermions, and most chemistry involves the leptons associated with an atom, rather than direct interaction of their nuclei, the fact that the nucleons were spin-3/2 rather than spin-1/2 would probably mostly affect the stability of certain elements. Instead of many isotypes of multi-baryon elements in our world that differ by number of neutrons, the variation among atoms with the same number of nucleons would depend on the extent to which the spin-3/2 nucleons had aligned or not aligned spins (and there would be only one kind of pseudo-hydrogen).
The mix of elements by atomic number might be different, however, depending mostly upon what the analog to the pion (which has a mass of less than 1/6th of the proton) which transmits the non-fundamental nuclear binding force in our world looked like. The mesons which have only second generation quark content are the 3 spin-0 pseudoscalar: charmed eta meson (mass ca. 3.0 GeV and no electric charge) and strange D meson (mass ca. 2.0 GeV and charge +1), the anti-strange D meson (mass ca. 2.0 GeV and charge -1); the spin-1 vector mesons: the Phi meson (electrically neutral and about 1.0 GeV), the electrically neutral J/Psi meson (electrically neutral and about 3.1 GeV), the vector strange D meson (electrically +1 charged and about 2.1 GeV) and the vector anti-strange D meson (electrically -1 charged and about 2.1 GeV).
Since all of these mesons have greater masses relative to the uncharmed omega meson that would be the predominant nucleon in this world than the mass of the pion relative to nucleons in our world, the nuclear binding force carrier boson in a second generation quark only world would be much heavier than in our world relative to the hadrons bound in a nuclei which would tend to make heavier elements in the periodic table much less stable than they are in our world (in part because heavier carrier bosons have a shorter range and in part because heavier carrier bosons relative to the hadrons would be produced less frequently).
Also, the Phi meson is by far the lightest, so it would be the most obvious primary carrier for the non-fundamental nuclear binding force (and would have a very long lifetime eventually decaying to photons or to muon and anti-muon pairs), but since it is a vector meson, rather than a pseudo-scalar meson, this would significantly alter the character of the interaction, making it more like a spin-1 weak force boson or gluon with dynamical mass, and less like the pseudo-scalar meson which acts as the carrier boson in our world. In particular, a spin-1 nuclear binding force meson would probably give rise to more internal structure on pseudo-atom nuclei than in nuclei of our world where the nuclear binding force is carried by a spin-0 pseudoscalar meson. The nuclei of pseudo-atoms might have something of a molecular-like structure.
And keep in mind that about
99.93% of stuff in the universe is made up of only ten chemical elements (just four, Hydrogen, Helium, Oxygen and Carbon) make up more than 99%. Two of those ten chemical elements (helium and neon) moreover, are chemically inert. Our world only seems complex chemically because we live in an extremely atypical little corner of it.
Basically, in a second generation only world, the chemical equivalent of hydrogen would make up a much larger percentage of all matter relative to pseudo-helium and other heavier pseudo-elements, than in our world. And, since pseudo-helium in this world, like helium in our world, would be chemically inert, the share of molecules that were simple two atom hydrogen molecules would be much greater, and most other molecules would be one or more pseudo-hydrogen atoms bound to a single heavier than pseudo-helium element (e.g. pseudo-water and pseudo-methane).
In particular, nuclear fusion, which powers stars and arises from the reduced binding energy per nucleon in heavier elements relative to lighter elements, would be far less potent in this world than in our own. This would mean that stars would produce fewer heavy elements, that stars would emit fewer photons relative to their mass, and that stars would collapse into black holes are far lower luminosities. Supernovas would be less energetic. My intuition isn't clear on the question of whether these dimmer stars would burn out faster or slower, but it certainly isn't particularly likely that they would live the same length of time as stars of comparable mass in our own world.
This doesn't mean that there couldn't be enclaves in this universe of heavier elements in planets, just as the tiny percentage of non-light elements in the universe are concentrated in places like terrestrial planets like Earth, and they could have interesting and Earth-like chemistry. But, with less stable heavy elements, this world wouldn't be one that is almost indistinguishable from our own.
A third-generation particle only world where top quarks could decay into bottom quarks would be quite similar. A third-generation particle only world where top quarks were stable, in contrast, might have more heavy elements because pion equivalents containing only bottom quarks and anti-bottom quarks would be light relative to hadrons containing any top quarks.
