Why calculate negative frequencies of DFT?
Mainly because its easier. The FFT is a specific algorithm to calculate the DFT. However, it only works if you calculate ALL frequencies (regardless if you want them or not). It takes in N complex values and it spits out N complex values. So the FFT can be used to evaluate the your first equation but not your second. 

In your example, that's a trivial difference; but for larger FFT sizes this makes a substantial difference. It's a lot faster to calculate the values at negative frequencies with FFT and simply throw them away then calculating only the values at positive frequencies with a different algorithm.

This being said, there are ways for making use of the redundancy:


You can pack two real signals into a single complex, do a single complex FFT and split this into odd and even parts to get the DFTs of both input signals
There is a way to do a N point real DFT using N/2 point complex FFT. It just needs one extra stage of "unwinding" the result.