BLDC motor current sensing for load torque
How battery voltage discharge can affect motor current measurement
  depending on the same load... Does it will affect my current
  reading and the current will increase?


Torque is directly proportional to current, independent of battery voltage. 


  Could I sense the BLDC motor current directly on the battery high side
  or do I need to sense it on all three the motor phases?


Yes, you can sense battery current on the high (or low) side. In a battery powered application sensing the low side may be easier. Sensing phase current could be more accurate, but is also more difficult.


  Is it enough to sense only current to create closed loop torque
  control, or do I need some additional parameter for torque control
  based on the different battery discharge levels?


To control torque you must use PWM. If you sense battery current then the motor current will be higher at lower PWM ratio, by the same proportion eg. if PWM ratio is 50% then motor current will be double battery current. This occurs because during PWM 'off' time the controller recirculates back-emf current back into the motor. This current never reaches the battery leads, so to get the true motor current you must multiply your battery current measurement by the inverse of PWM ratio.