How to deal with the mirrored components of a FFT? And another question
It depends on your spectrum analyzer.

If you want to analyze a one channel real baseband signal, the mirrored frequencies are redundant.  you should keep the first N/2+1 points.

For communications signals, there can be an I and Q (two reals as a single complex), you want to keep them all.  the frequencies will not be (in general) symmetric.

If you want to calculate the cross spectrum of two reals, there will be mirrored response, but if you want to do something like taking the inverse of the cross spectrum, you should keep both, but only display half.

It really depends on the actual application. There really isn't a right or wrong way. There are conventions and exceptions. There are niche applications. Spectrum analyzers that are commercial standalone test equipment, usually have lots of buttons, lots of modes. 

for your second question, this is what spectral aliasing looks like if you don't apply  an antialiasing filter prior to the DFT.