Yes that’s correct if it’s a new job. The education department needs to approve your employment so it needs to have been done within three months. However , after that if you stay with that employer you do not have to keep renewing until the next contract renewal.
Depends on the job, for example if its working at a cram school then the police check is good for 6 months according to the Ministry of Labor. If its the same school then no need for a police check again when renew.