1. what is the correct procedure of paying final settlement in mutual consent?
There is no hared and fast rule about this. In fact there is no provision in law for such settlements. It is a mutual agreement based on the compromise and understanding while the negotiations went on.
There is nothing wrong in telling them that you will pay the agreed amount at the time of judgement and for confirmation you may purchase a DD for the amount and may even give them a photocopy of it as an assurance.
2. is their demand of asking to pay beforehand justified?
Since there is no rule or law binding such agreements, there is no question of justification by any side.
It is purely on the basis of mutual faith.