Is there any law of quantisation for mass, just like for charge?

One simple rule to think here is that electric charge is a scalar quantity with magnitude. It has positive and negative signs, depending on the direction it is forced to move in an electric field. Mass is always positive. So when you add mass, it never cancels out or becomes zero. Also do consider that a charge can never exist when there is no mass. In calculations, you must remember not to take in the mass but just the charge itself. 

No, there may be some charges inside the system. The overall charge is zero because the total positive charge will cancel out the total negative charge inside. This directly follows the principle of additivity of charge, where the positive and negative charges are the algebraic sum of each other. 

One rule to think here is that electric charge is a scalar quantity. It has magnitude. It has both positive and negative signs, based on the direction it is forced to move in an electric field. Mass is always positive, on the other hand. So when you add mass, it never becomes zero. 

No, the system can still have charges inside. The overall charge is zero only because the total positive charge will cancel out the total negative charge inside. This directly follows additivity of charge rule. The positive and negative charges are the algebraic sum of each other. 

When two or more individual charges are present in a system, the total charge will be an algebraic sum of all individual charges and not the vector sum. Therefore, an electric charge is considered as a scalar quantity.