Unexpected frequency components found in Periodogram
First of all, if you use a logarithmic graph, it is customary to use dB. In order to convert a FFT in decibels, we use 20*log10, not log10.

Second of all, the "unwanted" frequencies are 10^-30 times smaller than the wanted frequencies, i.e. they are insignificant. . The unwanted frequencies are most likely due to the limited precision of floating-point arithmetic and not due to the sampling process. 

https://en.wikipedia.org/wiki/Floating-point_arithmetic

That should answer your second point too...