What happens when I use Come to the Altar and sacrifice my Living mummy. What happens first? Do I first resolve the action of CttA and put a character into play and then resolve the forced response from Living Mummy putting my just played character also in the discard pile?

Same goes for using Shocking transformation with Living Mummy

You resolve the Action first then resolve any Response effects including Forced Response (s).

So Living Mummy would wound the newly arrived character as it will be in play before the living mummy can trigger his ability.