Questions about MP3 bitrate conversions
How exactly does the compression take place behind the simple ffmpeg commands?

MP3 is a perceptual coder. It creates a time/frequency representation and quantizes in the frequency domain primarily by calculating the spectral and temporal  masking thresholds of human auditory perception. It maximizes the masking of the quantization noise to minimize the audibility of the quantization noise.

Are spectrograms the best way to analyse the change in bitrates of an audio file?

No. Depends a bit on what you want to analyze, but a spectrogram doesn't correlate well with the audibility of compression artifacts. Most of problematic artifacts are temporal anyway, the loss of a few high frequencies is a comparatively smaller perceptual problem.

The STFT used, gives us the phase and the amplitude from which we take the magnitude and plot the spectrogram. So how different can I expect the phases of the higher and lower bitrates to be?

Phase will be fairly close for all spectral components that are above the masking threshold. Not so much for low energy components.

Is it possible to somehow recover the lost frequencies because ffmpeg based bitrate upscaling doesn't produce accurate results on the Spectrogram

No. If that would be useful in any way, the original inventors of MP3 (or AAC, DOlby, DTS, WMA, etc.) , would have included it in the codec. Many people have tried but it turns out the harm outweighs the benefit.