Rubik's cube solver (help required for hardware)
1) Is there a platform such as arduino which can accommodate 6 stepper motors.


Yes. Each motor driver will require one pulse signal and one direction signal. You will need 12 outputs.


  2) Do I need 6 drivers for 6 stepper motors


Yes.


  3) Which stepper motors should I use?


That's a product recommendation question and is off-topic.


  4) What is estimate costs for that?


That's your job!