Does animal fat and meat contributes differently to our body?
Have you read the Wikipedia article on Fat?  It has literally all of the answers you are seeking.  However...

1: Every single thing is bad for you if you get too much of it.  Animal fat, sugar, water, you can over-do it on anything.  Fat can be bad for you but it is definitely required.  In particular, "Vitamins A, D, E, and K are fat-soluble, meaning they can only be digested, absorbed, and transported in conjunction with fats. Fats are also sources of essential fatty acids, an important dietary requirement."  Fat does a host of other things, as mentioned in the article.

2: Fat is indeed a concentrated store of energy, but there's more to life than just energy.  Protein, like other nutrients, are required for life.  Protein happens to be particularly important, providing the structure for all our cells, is required for muscle growth, and provides amino acids, which are needed to make other proteins in your body.  Proteins are used as hormones, cellular messengers, immune receptors, so on and so forth.  You couldn't live without protein.  Energy, while concentrated in fat, could come from elsewhere.

3: It worked just the same for them as for us.  I will add, however, that our ancestors did not necessarily use every part from every animal, and these days, we actually use every part of many animals, just not always for food.

4: Sure?  Energy is good, but protein is probably better.  "Athlete" is a vague term, as strength-training and endurance activities are very different from each other.  Protein is probably better for strength, carbohydrates are probably better for endurance.  But you can ask over at https://fitness.stackexchange.com/.