Hi Conor
Please do not apologize for your questions. We all have them from time to time, and it is those questions and their answers that give these forums a reason to exist. Without those questions and their answers, these forums very quickly degenerate into the mindless "my keyboard is bigger, better, more expensive than your keyboard" one-upmanship sessions that come from not having enough real issues to discuss. AND . . . all else said, the Casio manuals could use a good rewrite.
NOW . . . I will take your latest questions in reverse order.
INITIALIZATION is a single button (or menu item) RESET of a group of parameters or settings to known default (factory) values at the start of design of a new voice, or rhythm, or effect, or song, etc, so that you are not having to continually change a lot of ad hoc settings carried over from previous sessions. That is, with initialization, it is safe for you to assume that unless/until you have changed some parameter or setting that it is at some initial factory value. For instance, if you are designing a new tone, you know that, for that voice, reverb and chorus are off until you turn them on, and that once you turn them on, their amounts and delays are at zero until you increase them, etc, etc. Initialization of the entire keyboard resets everything to factory defaults and erases everything in User Memory - songs, tones, rhythms, registrations, etc, so you always want to make sure you have User items backed up before you do a general keyboard initialization.
I am not certain what you mean by saving the Mixer settings, and hopefully, if I am wrong on this, one of the other users out there, who knows better, will correct me on this, but I do not believe it is possible to save the Mixer settings as a group of settings unto themselves. If that WERE possible, then there would need to be a way to "name and save" them, such as MIXER-1, MIXER-2, MIXER-3, etc, for later recall, but there are no provisions for that. Normally, (some, most, all, ?) Mixer settings are recorded as part of a SONG SEQUENCER song, or PATTERN SEQUENCER pattern, or as part of a REGISTRATION. The parameter tables in the manual will show you what all gets saved with the different functions. If you have a preferred set of Mixer settings, you could save them as part of a REGISTRATION, but the problem with that is some REGISTRATION settings get inadvertently changed when other items in the REGISTRATION get changed manually. For instance, you can set up a REGISTRATION with a particular TONE and a particular RHYTHM, then change either one during live play without affecting the other (as long as you don't use One Touch Settings), but if you have declared a Minus-One-Octave offset as part of the REGISTRATION, it will get cancelled as soon as you change the TONE in real time. In that case, you would need to set up a REGISTRATION for each TONE and both having a Minus-One-Octave offset, and switch TONES with the REGISTRATIONS rather than switching TONES discretely in real time. So, I am not certain how useful REGISTRATIONS strictly for Mixer settings would be. It might be worth a try though. AS far as Mixer settings for the SONG SEQUENCER, I suppose it should be possible that, if you have, say, three different sets of settings that you use over and over, you should be able to create (record) three different two-empty-bar songs, each with its own Mixer settings and save them as Template-1, Template-2, and Template-3. Then load them as needed, immediately rename them, so as not to over-write your boilerplate templates, make any necessary minor Mixer setting changes, and proceed with your new recording. You just have to remember that you can only have five songs in User Memory at anyone one time. The sooner you exhaust that, the sooner you have to start shuffling things back and forth between User Memory and SD Card storage, but it still sounds like an interesting prospect. PC DAW users have been designing and storing quick set up templates on their hard drives for years. Here, we are just talking about doing the same thing on the keyboard with its SONG SEQUENCER set up.