Electronic – How to obtain 12 V DC, 3.3 V DC and 5 V DC from a single Li-ion cell

There are multiple techniques with various tradeoffs between them. If you search the literature for "battery fuel gauge" you should come up with a number of techniques.

Note that you can actually estimate SOC from voltage, provided the load is zero and the battery has been allowed to rest in this state for a while (hours). This relationship is not linear though, so you need to characterize the particular battery you're using and use some sort of a lookup table. Your voltage measurements also need fairly high precision and accuracy.

A more reliable method is coulomb counting, during both the charge and discharge cycles. This will provide a very accurate assessment of the amount of stored energy, and you only need to subtract out any self-discharge (which should be very small). However, measuring the current is a bit more costly, in terms of size, power loss, and $. And due to the self-discharge rate being variable and unknown, this technique needs periodic calibration or it will start to drift off from the true reading. This is typically done by resetting the high and low points after a full charge or full discharge respectively.

You can also combine these techniques (and possibly others) for even more accuracy and to mitigate drift effects.

This is how one manufacturer of battery fuel gauge ICs implements it: https://www.maximintegrated.com/en/design/partners-and-technology/design-technology/modelgauge-battery-fuel-gauge-technology.html

The residual energy in the AA cells is not valuable, the fact that the cells are portable, safe and cheap is.

That means using them for battery charging, that could use wall power, which is a few orders of magnitude more powerful and several orders cheaper (than full price batteries at least), is a total inversion of values and waste of time.

I too had access to a stream of partially used discarded AAs, and was able to make a steady donation of them to a charity serving the visually impaired, for use in their handheld illuminated magnifiers. They were perfectly matched to this service, as premature exhaustion of the cell was of no consequence to the users, especially given that they were free. The key here was a reliable stream, which allowed the service to abstain from buying full price cells.

If you insist on trying to recover the energy (I'm trying to imagine a desert island or post-apocalyptic situation where you don't have mains power, but do have the theatre's discarded AAs), then a) mixing AAs of a different charge state will result in premature run-down of the string of cells and loss of efficiency rather than damage (as long as you remove flat cells promptly) b) inverting up from a single cell is the least efficient way to do it, better to use a higher voltage and c) 18650 can be charged by a low current without reduction in life, as long as you continue to observe the normal charging envelope.