Will enough energy in a an appropriate environment, such as pressure, break apart any bonds matter may take, say for instance carbon dioxide. Also, if that’s true or not, is there a point in that regard in which elements themselves will begin to break apart, such as just carbon, giving enough energy in an appropriate environment again.

Basically yes, though it's also going to be high temperature that is another relevant environmental factor. You'd need both to ensure the combination of enough energy and collision frequency.

 

For elements, you can ionize them, forming a plasma. Breaking apart the nucleus is possibe, but it's more common to have fusion because most large high-temperature systems (i.e. stars) have mostly light elements in them, and fusion is energetically favorable.