Checking yesterday, passport processing is currently taking 3 to 4 weeks provided all documentation is in order when the application is submitted. Expedited processing (additional charge) takes 2 weeks (10 business days) for a serious additional charge.
To obtain a passport for a minor child, you need to prove that you have a court order of sole custody, or else both parents must sign the passport application. If the child will not be traveling with BOTH parents, you need court order of sole custody & permission of the court to remove the child from the country, or you need proof that the other parent is deceased, or you need verifiable written authorization from the other parent to take the child from Place to Place from Date to Date.
Rates are set by the government and you have no recourse but to pay the mandated fees with your application. You also have to pay for the passport photos, pay expedited processing (if you want that), and pay return shipping for the passports with expedited processing (e.g. Fed Ex charges). If you do not have the appropriate court orders, etc, you will need to pay court costs and legal fees, or if need letter of authorization, you need to pay for notary fees.