Married Filing Separately and First Child

Hello,

My wife and I do married filing separately as it is advantageous since she is on an income-based repayment plan for her student loans. We live together and are having our first child in October. We had a few questions and wanted to talk with an expert about our general situation and also the below specific questions:

1) Can we choose who claims the child on their tax return? 

2) Based on #1 above, does it matter which of our health insurance the child is under?

3) Based on #1 and #2 above, can one or both of us establish a Dependent Care FSA through our employer to pay for child expenses?

4) We believe we can still claim part of the child tax credit due to our married filing separately status but wanted to confirm. 

Thank you!