There have been very few peanuts that live. The longest that I have had one live is 4 weeks old. They usually die by about 4 days old because they can't digest food. I usually just put them down at birth now. Better than starving to death.
What breed are they? The little one looks like a peanut to me. If they are a dwarf breed such as mini-rex and both parents carry the dwarfing gene than you will get 25% peanuts also known as double dwarfs and they will not live.
They will usually start exploring at about 2 weeks old. I take the nestbox away at 3 weeks in the summertime. I give them a piece of board to sit on and plenty of hay.
It depends on the breed but most does should not be fed more than what they were eating before they were bred especially if they are a smaller breed. A smaller breed sometimes has smaller litters and the kits get too big if the doe is free fed. They also put on too much weight being free fed and...