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She’s a shrew. That’s why I always jokingly call her Katherine.
About Emily...

It was Richard's hard work that built the beautiful house he and her lived in, and when they separated, she had the nerve to make Richard live in the pool house.

Leading up to their separation, the show made a subtle point that Emily, like a lot of spouses of her time, contributed quite a lot to Richard's success in the form of managing his social calendar, (building social capital) decorating their home and maintaining his wardrobe (signaling their status to their peers) and doing all the other (unpaid, and often unvalued) domestic work to allow Richard to focus on his work. And while Richard did benefit from those things, he was completely oblivious to them, and thus felt no appreciation or gratitude towards Emily for her labor. 
You can argue she's not a good person, but 'immature' seems a poor fit.