Do we consume DNA and proteins of other organisms?
Yes, we are eating the DNA and proteins of an organism when we consume and digest its cells. 

However, consuming the "raw flesh of hen or fish" in your example would not cause harm strictly due to the consumption of their DNA.

DNA has two basic components - a structural 'backbone' of alternating phosphate molecules and sugar molecules (called deoxyribose) and a nitrogenous base (a group of atoms that includes nitrogen) on each backbone unit. These basic components of DNA are common to the all organisms that contain DNA, like the hen and fish in your example.

When you digest the DNA in your food, specialized molecules (in this case nuclease, which digests DNA) produced by your digestive system break down DNA into its components. These components can then be rearranged to create 'human' molecules, such as human DNA.

In other words, you don't integrate fish or hen DNA directly into your human DNA; you break it down and rearrange it into human DNA, then integrate it into your body.

To answer your second question, we get energy by breaking down the proteins, carbohydrates, and fats (known as macromolecules) of other organisms. Much like DNA, these three macromolecules are broken down into their singular components. For carbohydrates, this would be glucose.

These components are then further broken down into energy (as a force) which powers cellular machinery to 'recharge' ADP (the 'discharged' version of ATP) back into ATP, a specialized molecule which can 'bring energy' to other parts of the body to power your body's processes.

ATP is not energy per se - energy is defined as the "capacity to cause change" (Campbell Biology, pg. 160) and do work. Energy is therefore a property rather than a distinct molecule.

Summary:


When we consume DNA of organisms, it does not inherently damage us as we break it down and rearrange it into completely 'humanized' molecules.
Glucose and ATP aren't energy per se - they are just molecules that contain lots of energy. ATP 'brings energy' to other parts of the body; glucose is broken down into energy to 'recharge' ATP. Energy is not a molecule - it is a property.