(a) Yes. (b) No.
Unfortunately, this question rarely comes up with married couples but comes up again and again with divorced and separated couples. The default socially and seemingly in our tax system is that the mother gets to claim the children. However, the reality is that either parent can make the claims.
This is often a point of contention between couples with joint custody. I have seem many disputes over who gets to claim the children as tax deductions and have also seem cases where both parents tried to make the same claim and the result was chaos and audits and large arrears tax bills.
Enjoy fatherhood and if you are separated don’t think of the kids as tax deductions. Enjoy them.