First, about "why we have negative frequencies and which one is copied"...
In a real signal, there is no difference between a negative frequency and a positive frequency in this sampling context. Consider that you have captured 1000 samples that you know contains a single sine wave. You analyze it and find it contains exactly ten cycles of a sine wave. Knowing the sample rate, you determine that it's a 100 Hz sine wave. Now analyze the capture from the other direction—end to start. You would also find that it's 100 Hz, but since we moved backwards through time, it would be -100 Hz.
That is, if you choose to plot negative frequencies, they will always be a mirror of the positive frequencies. You will never find -80 Hz without 80 Hz, for instance. There is never a choice of "which on is copied". If you find somehow that there is a -80 Hz component, you can be sure there is an 80 Hz component.
Sampling is a type of amplitude modulation (AM). Pulse Amplitude Modulation (PAM) in the analog domain, Pulse Code Modulation (PCM) in the digital domain. In the analog domain, it's a multiplication of the signal with an impulse train at the sampling frequency. Looking at just the first harmonic of the impulse train, it's a sinusoid (cosine in particular) at the sampling frequency. In your case this means you are multiplying your signal by 800 MHz (as well as its pulse harmonics, but we need only consider the closest one here).
Amplitude modulation produces sum and difference frequencies, but for this problem we only care about the difference frequencies at the moment. That means if your signal has a 120 MHz first harmonic, sampling will also produce 800-120 HMz = 780 Mhz. And the fourth harmonic of your signal, 480 MHzHz, would produce a new component of 800-480 = 320 MHz. Note that this is exactly the same answer as your "-4*120M+800M". Except that I came up with my answer without considering negative frequencies (800M minus 4x120M).
To summarize, negative frequencies always have positive counterparts. And there is no particular need to consider negative frequencies with what you are looking at—it seems it just adds confusion, but may make more sense in context of the document you refer to.