The minimum length of the beard is what can be seen as hair on the face by people who see you from usual distance of conversation, no matter how short it might be.
Yes it is not only allowed but even recommended (Mustahab) to dye the gray beard and gray hair with any color which suits you. Some non Shia Shaikhs claim that black color should not be used. This claim is not authentic and has no evidence. According to authentic Hadeeths, using any color which suits you is allowed.
Applying any material on the skin or scalp is allowed even if it is Najis material as far as you clean the area before performing Wudhu or Salaah. Synthetic alcohol is not Najis according to many Maraaje'. Some scholars say that any intoxicant alcohol is Najis. Nevertheless, whether it is Najis or not Najis, it is allowed to apply it on the body or scalp.
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