I don't believe Stephanie took Ethan with the intention of murdering him. I think Stephanie took him for two main reasons:
1). Because she could. Because it was a way to hurt Joe. Stephanie getting Ethan for the summer, especially against Joe's greatest fears and wishes- that he made public in a court document- was a 'win' for Stephanie, in Stephanie's mind. It was a big "HA, HA, HA" directed at Joe.
2). Because it was a way for her to get more money. When she thought of Ethan, she thought of dollar signs. I believe she was planning on hitting Joe up for 'child support' (whether official or under the table) during Ethan's summer stay. I believe she was also going to try and get all the government assistance (food stamps, etc., etc.) that she possibly could, and it's much easier to get that state aid if you have a young child.
~ I also believe that when it came time to return Ethan to Joe at the end of Ethan's summer stay, she was going to refuse- unless and until Joe gave her some kind of cash pay-off to return Ethan. In effect, she was planning on making Joe buy back his son. She was going 'to sell' Ethan again.
~ I think there is also a chance that Stephanie had a tiny afterthought reason to take Ethan: to show her 'man', Nathan, what a great 'mommy' she is. Kind of a 'look how great I am, I'm such a good mom and woman' type thing. Like how we are told she presented herself to her former co-workers while she was still married to Joe.
But something went wrong. Something angered her. It was probably something innocuous that set her off- like Ethan referring to Joe's fiancee Becky, or maybe Joe's other ex-wife Michelle, as 'Mommy' in Stephanie's presence. Ethan was only four years old, he was in a new, unfamiliar place and situation with two people he barely knew (who were probably both using drugs, imo), and probably emotionally stressed, which is understandable. For some reason I keep thinking the word 'Mommy' came out of his mouth, and Stephanie was embarrassed in front of Nathan that Ethan didn't mean her.
I wouldn't be surprised if the physical abuse was actually started by Stephanie. She may have slapped Ethan, Ethan would have started crying, and then Nathan, wanting to show his support to, and solidarity with Stephanie, told her to stand back, that he'll "take care of this".
Being the macho guy that Nathan viewed himself as, he took on the roll of 'firm disciplinarian', and took a little 4 year old child into the bedroom, and beat the daylights out of him.
From that point on, the more Ethan cried (and Ethan would have naturally been hysterical after the first beating) the more it would enrage Nathan, and in turn Stephanie.
At some point, they completely stopped seeing Ethan as a child, (if they ever did to begin with), and he simply became their personal little punching bag.