The universe appears to have a finite age in the Big Bang model and can be calculated to give an approximate value. Particles however, do not seem to have any correlation with time. The lifespan of a fundamental particle, without colliding with its anti-particle through annihilation, appears to be infinite.
Yes, some of the elemental particles in our body (electrons, quarks, …) were actually popped into existence along with the universe during the first fraction of time of the Big Bang, and remained still until today. So whatever or wherever these particles came from (assuming they do), they carried with them information that is older than the universe itself.
If I am not mistaken, accordingly to the Big Bang theory, all particules were created at the same time. Not before or after, or few of them but all of them at the same time.
No new electrons/quarks/… were created since then.
Photons are an exception (at least, in as much as that they are a particle), and you can make new particls from energy, but definitely there’s a limit to how old a particle can be. No particle is older than the universe (as far as the big bang is concerned).
The universe in the Big Bang theory does have an age (approximately 13.7 billion years), so in a way, the Big Bang theory does not explain the origin of the universe, but rather the state of the universe 13.7 billion years ago. With that said, it is the limit of the universe, not the particles. The age of a particle could be infinity.
Except the big bang is the start of both space and time, so nothing in the universe could be older than the universe because there’s no time to speak of (not to mention the space for it to exist).
The moment after the big bang is called the Planck epoch. I just learned this from Wikipedia “In this stage, the characteristic scale length of the universe was the Planck length, 1.6×10−35 m, and consequently had a temperature of approximately 1032 degrees Celsius. Even the very concept of a particle breaks down in these conditions. A proper understanding of this period awaits the development of a theory of quantum gravity.”. I don’t really understand this, but it seems the early universe wasn’t conducive to particles. Even if it was, they wouldn’t be atoms. They’d just be quarks.
All of our physics breaks down at the singularity before the big bang, so assuming quarks that are around today existed then is just that, an assumption.