If I am filing an I-130 as a U.S. citizen to bring my spouse to the U.S., do I have to file separate I-130s (and pay a separate filing fee) for our minor foreign-born children?

The answer is yes.  A separate I-130 must be filed for your spouse and for each foreign-born minor child.  The only exception would be is if your foreign-born child is actually a U.S. citizen by operation of law.  Those scenarios are complicated and infrequent.  The general rule of thumb, therefore, is that a separate I-130 should be filed with the U.S. Citizenship and Immigration Service for each child.