At the beginning of the episode, Zack and Kelly say they have been going steady for 68 days, 14 hours, 37/38 minutes. Yet just one episode prior, Zack asked Kelly to go steady and then fell for the school nurse. At the end, they weren't going steady, they were still dating other people.
Zack's parents don't appear to have a problem with the unusual behavior of his Principal Mr. Belding coming over to Zack's house just to hang out after school. No respectable High School Principal would do such a strange thing.
After Mrs. Belding kicks him out of the house, Mr. Belding apparently lives at Bayside for a period because he has nowhere else to live, which would not only be illegal, but also very risky behavior on his part. He could've stayed with a relative, a friend or at a hotel.