Hard to tell from the pictures but if she has very soft short fur then she may be rex or rex mix or mini rex or mini rex mix. if around 4 pounds then mini rex or mini rex mix if at or above 10.5 then rex mix.
First I would separate each rabbit from each other. so the mom doesn't get stressed out and the other one might kill them. Also, you shouldn't be giving them that big of a carrot, they are high in sugar and 1 or 2 baby carrots each is all they should have and it should not be the main diet, pellets, and bunny safe veggies and small amounts of treats. Yes, you can move the babies get a cardboard box and fill it with straw and most likely the mom pulled fur so put that in there and place the babies in the box and put them in the cage with the mom. Also after the babies are weaned and the other bunny is put back in with her. Are they both female or are they female and male? they also need to be spayed and/or neutered to prevent more unexpected litters or fighting.