Calculation number and meaning of DFT
Plain DFT performs  N summations and N multiplications to give the amplitude of the frequency at that DFT bin but usually, we use FTT which is an algorithm for reduction of the number of multiplications.
Since you project your signal on the complex exponent, your output is complex too. There are different ways to represent the output. You can decompose the spectrum to amplitude and phase (Bode plot) or just ignore the phase and plot only the amplitude.

For a real input signal, the spectrum should be symmetric around zero and you can present the right plane only but for a complex input signal, you have to represent both sides of the spectrum. Regarding doubling the amplitude, I am not sure that it is mandatory but it is a common practice.