Syndrome apparently cleaned up the house before the Parrs came home because all of the damage Jack-Jack caused was missing. Of course, he wouldn't want them to get a hint that he was there. Fridge Funny then comes into play since he didn't think to ask Kari what caused the inside of the house to have scorch marks from fire and lasers. Syndrome is that egotistical.
Why Jack-Jack didn't attack Syndrome when he first arrived and slept soundly? Jack-Jack assumed the same thing as Kari, that Syndrome was just another babysitter. He was probably really tired due to his superpowered antics as well so he figured there was no reason to assume Syndrome was anyone bad. Not until later when he tried to take Jack-Jack away from his family.
Jack-Jack's reaction to his powers, as well as not understanding Kari's fear of them, could have something to do with him being a baby and not knowing how alarming this would be to someone else but if you think about it, there's no reason Jack-Jack would assume what he's doing is unusual. His whole family has powers as well as his father's best friend, who spends a lot of time with the family. He sees them use their abilities all the time. So Jack-Jack isn't doing anything strange, at least as far as he knows, and is probably excited to be like his family in that way.
Kari could have been killed a number of times from Jack-Jack's antics. There is the fire, the lasers, and him dropping stuff on her. Fortunately, he didn't activate his Super-Strength yet.
While it's played as a joke, Syndrome's casual, unassuming attitude when telling Kari that he's her replacement shows how good he is at lying and manipulation (though considering Kari's situation, he also got very lucky).
As far as we know, Kari was fortunate in her state of being in a way. She was really done with Jack-Jack that she didn't think twice to hand him off to Syndrome without a second's notice. Syndrome has proven to hurt anyone who gets in his way, so had Kari even shown a hint of resistance, nothing is stopping him from harming or even killing her.
