Yes, "Do you want to take care of my child" is a more common English formulation, though I wouldn't say that "Do you want to be taking care of my child" is wrong. (The version with "take care" was just accepted for me.)
Yes it's unnatural for sure. It's not wrong per se but I think it implies NOT doing something else.
"I'm sorry that you have to go but do you want to be taking care of my child?"
That seems like it might sound right, especially with some emphasis.