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.

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.