Yes, if you have a e-2 visa (English teaching visa), your spouse can enter korea with you and be given a f3 visa. The f3 visa does not allow your spouse to work, just staying with you during the time of your employment. Also you can bring your pet with you. For your apartment, some schools outside the big cities offer larger single apt or housing allownce. I suggest you ask potential employer for the apt for you and your family.

Good luck!