My question is that what are the rulings of mehr? Who gets to decide the amount of mehr and when and how much should it be given? What if a girl's father is adamant on asking for a big amount of mehr then what are the hadiths and rulings in this matter?

The mahr is the exclusive right of the wife and she is the one that has absolute right over it. She can demand whatever she wants and she can waive it whenever she wants.

My question is that what are the rulings of mehr? Who gets to decide the amount of mehr and when and how much should it be given? What if a girl's father is adamant on asking for a big amount of mehr then what are the hadiths and rulings in this matter?

The decision is for the bride, as the mehr is for her, not for her father, and according to hadiths, it is makrooh (not advisable) to ask for big amount (The lower, the better), and if the father is insisting on big amount, which stops his daughter getting married, then she can marry her self, without referring to him.