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I have tried all suggestions in comment but no good result. Finally, I added a 100 µF capacitor to VCC and GND of Arduino (5 V), and remove the 1 kΩ in pin 11.

Here's the update:

and the code:

  void setup()   
  {   
  Serial.begin(9600);   
   pinMode(11,OUTPUT);  
   digitalWrite(11,LOW); // Normally LOW inthis pin   
   pinMode(A0,INPUT);  // 10k Potentiometer   
  }   
  void loop()   
  {   
  int s=analogRead(A0); // 10k Potentiometer   
  int z=map(s,0,1024,0,255);   
  Serial.println(z);   
  analogWrite(11,z);   
  } 

Temp Result:

1. The motor is not too hot anymore.
2. I still have motor vibrations.
3. Motor speed cannot reach the max speed (I'm sure this is because of the vibration)

UPDATE 2

1. I change the thickness of the wire. I use all the same wires (28awg). Result = No heat.

2. I planned to remove the 10Ω to reach the MAX Throttle/speed. But, when the arduino was powered the motor started to spin in 6 seconds and stop - even though the potentiometer shows in min speed. Then I added 150Ω and It worked as expected !!! (based on this article).

By doing the two above, I finally didn't see the vibration!!!

Noted: I also try using 4.2 LiPO battery (8520/720 coreless DC motor can handle this based on its datasheet) and it spinned so fast (max throttle) without vibration and heat.

Time's to use this project for my DIY Arduino mini quadcopter!!! Thanks all!