The stepper driver you linked to would be much more suitable for a 2 amp motor. An Easydriver or even a BigEasydriver cannot provide enough current.
The 12v battery should be fine - at least while it is well charged. The 7Ah figure is a measure of its capacity, not its ability to provide current. It can probably provide 10 or 20 amps easily if required - but not for very long. You should also assume that the actual capacity of the battery is only 50% of the sticker value
However stepper motors work better when you use a higher voltage - the driver in your link can work with 40v. I suggest to think seriously about using 24v.
Also post links the the datasheets for your stepper motors. Nema 17 and Nema 23 only tell you the size of the from face (1.7 inches)
ephemerality:
My first question is that since it is a 2 phase motor, do I need to supply 4A and 4.4V to this motor?
No, the current is 2A - that's the max amplitude of the current in any winding when that winding is
fully driven (depends on the (micro)stepping position when the maximum is).
You don't supply voltage to the motor, the chopper driver provides current to the motor, the voltage is
dependent on the motor speed mostly.