Electrical – How is the fuse amp rating determined for a 12V AGM battery bank

There are really two questions here.

1. To calculate the run time you first need to estimate the amount of current being drawn by the inverter. If you assume an efficiency of 85% (your millage may vary) then you can calculate the current as
   ((120V / 24V) * 7A) / 0.85= 41A. This number is a bit on the high side since the voltage will initially start around 26V and drop down as you discharge the battery. The run time for 50% discharge is then calculate as a) (200 AH/41A) / 50% = 9.7 hrs. In case b) you have double the capacity so the runtime is increased by two.
2. You don't get any more charge discharge cycles by combing batteries in series. However, if you combine them in parallel then you would only be discharging the banks by half as much for a given a given amount of time. So even though your charge/discharge cycles wouldn't change, you would get twice as much work out of the batteries.

There is no problem with having a load drawing power while a charger is charging the batteries. The charger is pushing power into the system and the A/C is taking it out again. Most of the power will be going straight from the charger to the A/C, but that just means that your battery array will take longer to charge as it'll only be getting charged at 17A as opposed to 100A (100A-83A). This is the same as when you use your laptop or phone while it's plugged in.