You can clean the bed when the babies are about 2-3 weeks old and start leaving the nest. Lift the clean top part out, clean underneath and then put the clean part back.
Can you move the whole cage/hutch? If so then you could move them to your house about the same time. If you want to change the whole environment eg new cage then not until at least 3 weeks old when the babies are hopping around. Take the nest with them.
Do you enough room in your bedroom for accommodation for the mum and babies? Depending on how many she has had you may need quite a large pen. It would need to be dog/rabbit proof - remember babies will squeeze through very small gaps at that age. They'll also be noisy so you might not get much sleep!