NiMH cells start at about 1.5 V right when fully charged, drop to about 1.2 V most of their discharge life, and are pretty much empty at 900 mV. Stopping there is usually safe. 800 mV is where you definitely want to stop to avoid damage. There is so little energy left at that point that there is no benefit in draining the cell further anyway.
You may think therefore that your 10 cell pack can therefore be discharged to 9 or 8 V, but unfortunately not. There will always be some imbalance between cells. If you can measure individual cells, you can go until the lowest cell hits 800 mV, but then stop right away. The cell with the least capacity will get there first. Once it does, its internal resistance goes up and further current causes the voltage to drop rapidly, causing permanent damage.
10 NiMH cells really should not be put in series without a way to at least measure individual cells. If you designed the battery pack, then you need to fix this. If someone else did, then they are not trustworthy and it would be a good idea to dump them and find someone that knows what they are doing. With 10 cells, it's hard to pick a reasonable stopping point because possible imbalance between cells could be significant, especially after a few charge/discharge cycles. Maybe use 1.1 V average per cell, but this is really not a good way to deal with a 10-cell pack.
You will have the same problem with charging. You will have to use relatively low charge current, like maybe C/4 until you think the first cell is near full, then maybe a C/10 or so to trickle charge for a couple of hours so the other cells hopefully catch up. Again, the right answer is to not get yourself into this mess in the first place. Packs with this many cells need to have individual cells measured at least, and the best way is to have some charge balancing circuitry. This shunts some charge current around the full cells so that they don't get overcharged while the less full cells catch up. Of course this requires measuring individual cells to know when to enable the shunts per cell.
Again, work with someone or a company that actually knows what they are doing. This sort of thing is a lot more complex than it appears at first glance.
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.
Best Answer
The charging voltage will be 1.4 V -1.6 V per cell. They must be charged at constant current, not 1.2 V per cell.