Why Did Tate Kill the Exterminator?

Critter is frightened by what he sees, but Tate shows up, and in order to keep Critter from telling Ben and to protect Violet, he shoves Critter's hose into his mouth and causes him to ingest his own poisons, killing him.

Why does Tate not remember?

But Tate doesn't remember anything, so the story is left unfinished—an easy guess is that he didn't kill himself at school, as the Columbine shooters did, since he haunts The House, which we know only happens if you've died there.

Why did Tate kill Chad and Patrick?

Tate donned the suit and killed Patrick and Chad after they decided not to have a child, hoping that a new family would move in and have a child, which Nora could then have instead. Moira gave Tate the couple's gun, which Tate used to make it look like a murder-suicide.