じゃありません is conjugation of です. いません and あります are conjugations of verbs いる and ある. That’s why, you can use じゃありません in the same way you would use です. And, as you noticed yourself, いる and ある have their own rules of usage.
I don’t think there are any rules for pronunciation of じゃありません. If there were, Koichi would surely mention them. It’s just one of those things, like how せんぱい actually sounds like sempai.