Who should be the one giving money to MIL for taking care of baby

err, this depends on your financial situation. if you not giving in monthly basis, maybe you can consider giving hangbao (red packet)while birthday, bigger amount hangbao on CNY. or buy the food she likes or small gifts. well, taking care kids is not a easy job somehow show love n care that you appreciate her. make someone happy ya

Depends on your family. For us, both contributes, whoever earns more gives more.

Doesn't really matter since it's between both of you

it really depends