Can my rabbit run around my house 24 hours a day?

An untrained rabbit probably should be kept in a cage while you're not home to supervise and at night when you sleep. Rabbits are crepuscular, which means that generally they sleep during the day and during the night but are ready to play at dawn and at twilight. Be sure to let them out during the evening when you are home, and if possible, in the morning while you get ready for work. When your rabbit is better trained, and when your house (or the part that your rabbit will have access to) has been sufficiently bunny-proofed, your rabbit can be allowed free run of the home (or part of it) even when you are not home. The more room your rabbit has to run around in, the more delightful you will find her as a companion.