Why did he exiled Sita?
Why did Rama exile Sita?
The general reason is that in Rama Rajya, Ram used to provide everything to his people what they wanted. So he exiled Sita for the sake of people..
But every Generalgeneral seeing thing have some deep mystic connection to something. Similar is the case here. It was because Sita was cursed by a parrot to happen that. These parrots were near Valmiki Ashram hence they have mastered some verses of Ramayana. One time they went to Mithila and Sita heard about her story singing by them. This is in Padma Puran.
For your second question:
Why did he asked for a second test ?
Why did Rama ask Sita for a second test?
Again the general reason is same as before i.iee., as a King for the sake of people.
But this also comes under a mystic leela. This was done to depart earlier from earth to his eternal abode. At that time age was very high. For egE.g., Dasharath was more than 60000 years old when Ram was born. But it was necessary for Rama to apartdepart earlier as he came to rule only for 11000 years.
Hence Gods Leela are very mystic. If we go deeper and deeper we find more and more mystic...