I need some advice. I have read that interacting fields can lead to quanta annihilation? How often does this happen, and can bound state quanta annihilate or is it just free quanta?
There are no "quanta that make up carbon".
Particles in a carbon atom interact with each other and their environment all the time, quantum field theory describes those interaction with creation and annihilation operators. This is a mathematical tool. It is not like a carbon atom would vanish somehow (unless you shoot antiparticles on it).
what does that mean in terms of many particle systems? as they are made of particles, are the parts of these systems always being created and annihilated or is it just single particles that are annihilated? the ones not making up a carbon atom for exampleI think you ran into some misconception about QFT here.
In QFT, particles are created and annihilated all the time - but they are not like billard balls, it is just a description of their interaction. The total number of real electrons and valence quarks (the quarks which define the type of nucleus -> proton vs. neutron) is constant.