Stepper Motors Calculations
how much power are the stepper motors going to draw, is it calculated
  by the movement or per hour?


That depends on how you drive the steppers. You can do it two ways.

First, you can drive the motors continuously. If you do so, you calculate the power by the hour.

Alternatively, you can provide current only for a brief period every time you issue a step command. If you do this, you calculate by the movement.


  what are the calculations for these ratings in order to determine how
  long my batteries will last taking into consideration I have a 9W
  solar panel where it will recharge them?


Well, let's say you are drawing 2.8 amps per phase, two motors, continuously. Then your total current will be 2 x 2.8 x 2 = 11.2 A. A 9 W solar panel providing 3.7 watts will provide 9/3.7, or 2.4 amps (asssuming perfect charge efficiency, which won't happen). Then for every hour of operation the net charge will be 11.2 - 2.4 or 9.8 A-hr. So a 6 A-hr battery will last 6 / 9.8 hours of operation, or about 37 minutes.

But now let's say you won't drive the motors continuously. Let's assume that you only need to drive the motors for 1 second to produce a step, and the motor holding torque will keep the panel in position the rest of the time. Then the question becomes how often you need to step. Let's work with 1000 steps per hour. Then you will need (1000 / 3600) x 2 x 5.6, or 3.1 A-hr. Subtract 2.4 A-hr for charging, and the net charge required is 0.7 A-hr. A 6 A-hr battery will last 8.6 hours.

Note that all of these calculations are very rough, and don't take them as very accurate. There are all sorts of inaccuracies implied. I've mentioned that charge efficiency is never 100%. Battery capacity varies with load: the higher the discharge current the less the effective capacity. A pulsed current controller will have a section which draws current constantly, so some attention must be paid to that continuing loss. And the list goes on.


  is this battery pack enough to power the stepper motors or will I need
  a higher battery pack, or possibly lower rated stepper motors?


I have no faint idea. You have not remotely provided enough information. Do the motors have to be driven at full current? If run pulsed, is 1 second enough? Is it more than necessary? How many steps per hour are actually required? Is holding torque enough? 


  does the L293D driver IC work with these two stepper motors or is
  there another driver IC I would need to use?


Now that is easy to answer: the L293 is almost certainly inadequate. Read the data sheet and notice the third sentence in the description, "The L293D is designed to provide bidirectional drive currents of up to 600-mA at voltages from 4.5 V to 36 V." 2.8 amps is a good deal more than 600 mA, yes? 3.7 volts is rather less than 4.5 volts, yes?

Of course, no one says that you have to run your motors at 2.8 amps per phase, although perhaps you do. That will depend entirely on issues like your mounting and drive mechanisms and wind loading. If you assume (just for grins) a 1.2 volt drop on the L293, adding a 2.8 ohm resistor in series with each phase will produce 600 mA current. At 600 mA, total motor drain is 2.4 A, which just equals your nominal charge current of 2.4 A, so this will work just fine. Nominally. Ignoring all the pesky effects which I have mentioned earlier.