I am a MB "rookie", but as far as your question #2, MusicBee uses the BASS audio library (www.un4seen.com
), as does Aimp. Foobar uses a combination of customized ffmpeg libraries and its own core decoder. I don't have Media Monkey, but am 99% sure this is a BASS project also.
As far as audible differences between MB and Aimp, this could be due to different equalizer settings or use/non-use of replay gain, pre-amps, etc. Aimp also resamples EVERYTHING--while MB does only if selected. Whether that may have an audible effect on the sound is debatable and continues, but the prevailing theory appears to be "no", if the sampler is operating properly (Aimp uses the well-reviewed SoX resampler). But feel free to correct me folks!
As I don't use replay gain I can't answer your question #1 but I'm sure others will.
For #3, selectable on-the-fly normalization, you can make a post in the Wishlist forum for it to be considered by Steven and/or plugin developers.