This is quite the multipart and deep question. You seem to understand the basics but just in case, I’ll recommend this website as a good overview, albeit dated in terms of current ICs and BMS competitors.
http://liionbms.com/php/index.php
Chips
This is dependent on application. For small packs like the one you’ve drawn, there are a wide variety of chips available. For larger packs, Linear (LTC6803) and Maxim (MAX11081) are the two main competing suppliers of BMS solutions. They provide the most integrated solutions for multi-cell management. The main differences from smaller packs are the higher series voltages, daisy chain communication and noise immune signaling.
Techniques
In either case, voltage monitoring, temperature monitoring and active balancing tend to be the primary needs of a BMS. Other features such as redistribution tend to be less critical and often do not warrant the added cost and hardware.
Solutions
Unfortunately, even though nearly all BMS designs have the same goals, rarely is there an all in one solution. Solutions are based on number of cells, voltages (chemistry), tolerance for loss, communication method etc. These variables are not all standardized between one project and another. Furthermore, technologies keep changing. Linear is considering replacing CAN-BUS with shielded Ethernet. ADC accuracy is increasing. Sample rates are increasing.
The tried and true method is just simple active balancing. That is extra voltage on a cell is bled off resistively. Other than that, the rest of the system is a simple matter of reading all the cell voltages efficiently.
The Future
Eventually, we will see a more robust landscape with better standardization. Companies like Elithion or Nuvation are designing BMS solutions that are nearly drop in solutions. Chip designers are getting a better understanding of customers needs and have already sought to distinguish between the different types of li-ion needs based on different industries. This will mean that we’ll start only paying for features we need. Even battery cell manufacturers are standardizing cell types. For now though, any battery design remains quite customized.
A 12 volt "Smart" charger will deliver its full rated current until the battery's voltage reaches about 14.4 volts, then will reduce the charging current to maintain 14.4 volts. After some time, the charger will reduce the charging current to allow the voltage to drop to a float charge level.
If you try charging a 12 volt battery with a 24 volt charger, the charger will deliver its full rated current for ever, as the battery voltage will never get up to the 28.8 volts the charger is trying for. The battery will eventually boil dry and be destroyed.
Best Answer
It should be safe to use the same charger for both battery types.
AGM batteries are usually charged to a peak voltage of 2.4V per cell, which is 28.8V for a 12 cell 24V battery. LiFePO4 cells are fully charged at 3.6V per cell, which for an 8 cell '24V' battery is also 28.8V.
How to charge Lithium Iron Phosphate