Harmonic mixing in spectrum analyzers
In this case, it's all about the filters.  The mixer is followed by a relatively narrow bandpass filter.  So long as all the images can be filtered out, the spectrum analyzer will be able to unambiguously tune across the range.  If there was a point where two different input frequencies are mixed and end up at the same frequency as the first bandpass filter, then it's not possible to tune unabiguously.  The 1- and 2- lines show what input frequencies will be tuned to align with the first bandpass filter.  Since the lines never cross, then two different input frequencies will never be converted to the first IF at the same time.  

To your point 2, harmonic mixing is absolutely used to extend the range of the analyzer to higher frequencies.  However, it also has to be taken into consideration within each frequency band to ensure it won't cause any problems.  So, the answer is both: it can be useful, and it can be undesirable.  Just like many other physical phenonmena, including resistance, capacitance, and inductance.  Sometimes it's what you need, sometimes you get it for free and you have to deal with it (usually the case with parasitics).