Yes, only one tax return can set forth the child as a dependent.
Generally, the parent in the higher tax bracket should claim the child. For example, dermatologist is married to a school teacher. Dermatologist makes big taxable income and is in 35% tax bracket and school teacher is in 22% tax bracket. I would claim on the dermatologist return. The beauty of TurboTax is you can run it both ways and look at the result and make decisions.
You both can use you HSA to pay for medical expenses of the baby (your child is considered an eligible dependent for HSA purposes).
The law sources I reviewed says that the dependent "can" be claimed; it doesn't say must be claimed. My sense is that your FSA can pay for care of the baby even if your spouse claims the baby as a dependent.
