1. What is the maximum current can the energizer AA E91 produces?
2. What other methods can be used to reduce the number of batteries used?
3. How long does it take to drain all the current away from the AA E91 battery assuming my device runs continuously?
1) At least 5A (worst case, 10A typical) when fresh.
2) Err, using larger cells, like D-size, would help a lot and so will increasing the efficiency of the motor and drivetrain.
3) Your 'bot won't take 3A all the time, that's probably the peak drain. Besides that, several factors is mingling in, like temperature, whether running uphill, downhill or flat, how old the cells are, etc.
Remember, the 12V will only be 12V for a short time, as the capacity is rated down to 0.8V in your case (normally down to 0.9V). For most of the time it will be around 9.5V (give or take).
Don't fall pray of the usual confusion between A (Ampere), which is an instantaneous current value and Ah (Ampere Hour), which is a capacity rating.