Electrical – Is FM index affected by frequency multiplication

The answer to this fills whole chapters of some large textbooks[1], but to summarize a lengthy discussion involving Bessel functions of the first kind, the carrier component vanishes for M=2.405, 5.520, 8.653, 11.79, 14.93, ... Mathematically, these are the zeros of J₀(M).

[1]: Cook & Liff, Frequency Modulation Receivers, Prentice Hall, 1968, Fig 1-19, but you will find the same diagram in numerous texts.

With FM you get much more than two sideband frequencies. The sidebands you describe sound more like AM (amplitude modulation) sidebands. You may want to supply a link where you get this information from or re-check your information.

Next, in your final paragraph you might be getting confused. At any one moment in time the FM signal you describe will have a frequency that is somewhere between two limits. The limits are called the deviation meaning that's how much the carrier deviates +/- from its nominal centre frequency.

That deviation is nothing to do with the modulating signal's frequency but has everything to do with the modulating signal's amplitude. The bigger the peak amplitude the bigger the deviation from the nominal centre frequency.

If the PLL is designed correctly, and its low-pass filter permits the VCO to track the carrier within the deviation limits, the low-pass filtered signal (that feeds the VCO) will represent the signal that caused the original modulation i.e. the PLL is an FM demodulator.

You also mention "keeps re-locking" - this is not something that should happen in this type of circuit - the PLL will remain locked to the modulated carrier. There will be a slight error in the instantaneous lock because you need an error to drive the mechanism that tries to maintain lock but, this error will be small and this error is not regarded as the PLL losing-lock.

You might also be getting confused with what happens when the spectral content of FM is analyzed. Yes it has several components of frequency but these do not occur together at any one instant - the spectral content is a time averaged evaluation of what the carrier is doing - moving about following the amplitude of the input modulating signal.