Yes. You can go away for up to a year and come back to the same residency status. It's called a special re-entry permit. You need to fill out an "Embarkation Card for Re-Entrants" just before you go through immigration on departure. Don't forget to take your Zairyu Card.
If you will go abroad for more than a year, or you think you might end up staying away longer, you need to apply for a Re-Entry permit before you leave at your local immigration office.
No. The residency management system is only for where you live in Japan. I think you can just stay registered at your current address.