I don't think there is much to learn by running the EDF using the NiMH pack at 7.2V. If this is the same EDF that you have described in the other thread on ESC sizes, and you intend to run it on 3S lipo, (nominally 11.1 volts, fully charged 12.6 volts) then the only thing you will learn is whether the motor actually runs or not. You won't really be able to extrapolate forward to total current draw or thrust.
Yep - the XT60s are pretty standard and will work for this.
For the battery you are getting a 3S 2200mAh - try to get a 35-40C at minimum. They may be a little more expensive but a lower C, cheaper battery will need to be replaced sooner costing you the price of a new battery.