How do plants and animals add or remove carbon from the atmosphere

It is through respiration or gas exchange.

Maintaining homeostasis

Its not actually just one system if you look at it, there are various interdependent system that actually manages every movement of every system in order to maintain homeostasis, plainly every set function is a coordinated body system response. For example, the digestive system digests food because of the enzymes and the cells and organs present in its system but again, this is aided by the different hormones which is part of the endocrine system, interdependently. Thus eventually, the circulatory system is the organ system that intiates in dissemination of the different processed and intially metabolized nutrients in the digestive system and sent throughout the body. Incidentally, also the immune system acts as agent in again many systems to maintain the so-called homeostasis.

The plants use carbon