There is a sliding scale of difficulty here.
Is the Due fast enough for BLDC commutation?
In general, yes. If you can do 9000 RPM with an 8bit AVR, 10kRPM should be easily achievable, and you can probably afford to be a bit loose with your coding.http://www.instructables.com/id/BLDC-Motor-Control-with-Arduino-salvaged-HD-motor/
My question is: Is the Arduino Due able to do the above process in 6.25 micro seconds?
This I think is starting to push the limits, depending on how much calculation is performed, you can quickly eat up 5us and have not much left over.
If motor control was the only function, then it should still be doable, but you may need to do some optimizations and use whatever hardware assist is available. Will need to be interrupt driven.
check for that flag in main loop before any other processor intensive tasks (like lcd display, PID calculations etc.)
This is where you will probably run into difficulties. if you have the CPU 80% dedicated to driving the motor, the rest of the tasks have to be low-priority background stuff. There is no low-overhead way to assign priority to real-time tasks other than with interrupts.
With the right architecture, and some careful coding, you might be able to squeeze everything in. I would consider having a chip dedicated to motor control, and another for UI type functions.
Even with a detailed spec, it's not possible to give a definitive answer. The only way to find out for sure is to prototype it. If you write some representative control code and measure how much CPU it takes, you can get an idea how much is spare.