1. Correct
2. The controller measures the current and uses software to limit it based on the measurements. It doesn't know or care what battery you have. If the battery can't supply the current that the controller demands, you get voltage sag and shortened life of the battery.
3.You can't calculate the maximum current of the motor. If you use a larger controller to get more current, you get more torque, but, if you go too far, the motor burns. How much current you want to give it is a judgement. Some people fill their motors with oil for better cooling so they can go a long way higher on the current, but then there's a danger of stripping the gears unless you have a gearless one.
4. The aH rating is a measure of how much energy is in the battery. To know how much current it can give, you need to know the "C" rating. You multiply the "C" rating times the aH to get the max continuous current. A 1C 10aH Li-ion battery can give 1 x 10 amps. A 5aH 25C Lipo cang give 125 amps. The latest cells from Sony, Panasonic ans Samsung are now up to 3C, while as the cheaper ones are 1C. Normal LiFePo4 cells are 1C, but you can get special ones like A123 and Headway that can go very high.