How should I handle this puppy?

I have my first puppy. I've owned dogs, but she was a rescue and was already housebroken. I got my puppy last Sunday. She's a large breed dog and is 5 and a half weeks. I know she's young, but her mom died from birthing difficulties. She is doing great on potty training during the day. School is still in, so I have to have sleep. At night what can I do with her. She wines even when I sleep beside her. Mom says let her sleep outside of the cage, so I put up the baby gate at my door and wake up at about 4, but there is pee on the floor. During the day I take her out about every 20 minutes. If she's napping I take her out as soon as she gets up. She doesn't eat or drink 1 hour before bed. She goes potty when she is outside and I've been saying "Gotta go?" to take her out and "Do your business" when she's going potty. She's catching on quick. Mom is no help, so I'm trying. She's great when I'm here. How can I save my carpet at night??? Why is her potty a little runny? She eats purina pup.