1. No.
2. Yes, but - once it lapses you'll get a letter in the post asking you to renew. It's doubtful that an inspector will call out immediately. However you can eliminate that risk by simply renewing it and transferring it to your new home by calling into the post office with evidence.
3. This is covered by legislation. If you do not have a lease and are there less than 6 months, you must give 28 days notice. If you do not have a lease and have been there more than 6 months, you must give 36 days notice. If you do have a lease, the terms of the lease apply.
4. Only #3.