It depends on how you define childish, and also the age of the person in question, but in general: Yes.
Also, in case of adult women, if they speak in third person, to me it sounds あざとい(azatoi). I am sorry, I don't know how to phrase it in English without ruining the nuance. It's a mix of calculating, cunning, and a bit flirtatious? Either way, you also need to consider the tone of their voice, body language, and the contents itself. (Maybe if they want you to carry the shopping bag because it's a bit heavy for them, they might use this way of speaking.)
That being said, I do not have any real-life examples to back this up so take it with a grain of salt.
I see where you are coming from with that one. However, this acting doesn't often have that strong of a negative connotation as insincere seems to have. To me, it's a more light thing, a bit of playful acting, atleast for the scenario I mentioned in the previous comment.
