Your 6-cell battery holder will give you almost 9V when fully charged, which is probably too much for your servo to handle and higher voltages can have a negative impact on the motor lifetimes.
You will probably get a little less than 20 minutes, I would think closer to 15-18 minutes. The discharge rates will vary by the battery capacity and manufacturer; you should be able to find some generic discharge characteristics with a brief Internet search, and the manufacturer of the batteries you have might have a datasheet on their website. You might find the information in this forum thread helpful as well.
Please note that with such a high discharge rate, you will probably get a significant voltage drop with NiMH batteries. For high current draw applications like running multiple motors, I personally tend to prefer LiPo batteries for their power-to-weight ratio and high current discharge capabilities.
Regardless of the battery type you use, the batteries will only provide the current that your motors draw up to their maximum discharge rate (e.g. they will not “push” current through the system).
-Derrill